Breakthrough in the theory of stochastic differential equations and their simulation

Amin · March 10th, 2020, 3:49 pm

I was thinking of trying something on the lines of the following equation for the main update equation

[$] w(t+\Delta t)=\mu_w +[ \mu(t) \Delta t - .5 \frac{\partial^2 \mu(t) \Delta t}{\partial Z^2} H_2(Z)] + \sqrt{(([w(t)-.5 \frac{\partial^2 w(t)}{\partial Z^2} H_2(Z)]-\mu_w )^2 + (\Delta w)^2)}[$]
[$]+ .5 \sqrt{(( \frac{\partial^2 w(t)}{\partial Z^2})^2 + ( \frac{\partial^2 \mu(t) \Delta t}{\partial Z^2})^2)} H_2(Z) [$]

This is just a rough hypothesis at the moment but I will try this and its variants tomorrow. I decided to share it as a food for thought for friends so they could try their own variants if they find something appropriate in this idea. I will share the results with friends how it goes tomorrow.

Sorry friends for the last erroneous post though the basic ideas were right. It was very late at night when I wrote the post and I had not thought everything carefully. But Here I developed the ideas further and more correctly and I write them with equations.
First of all, we really have to find a decomposition of our w(t) in terms of hermite polynomials. I have tried to find a scheme to do that and the idea behind the scheme is that we put together the first order effects due to first derivative in terms of coefficients of Z, the first hermite polynomial and lump the rest of the effect into second hermite polynomials. We could have gone to third derivative but, as a start, I decided to keep the representation to first two hermite polynomials. Besides going to third hermite polynomial would require a sufficiently precise analytic calculation of second derivative of w with respect to Z which would not be required if we keep the representation to just two hermite polynomials. If we can work well with two hermite polymials, the loss of accuracy from not including the third hermite polynomial would probably be quite small.

Here is a typical representation of [$]w_n[$] associated with [$]Z_n[$] in our setup.

[$]w_n=c_0 + c_{n1} Z_n + c_{n2} ({Z_n}^2-1)[$]

We start finding this representation from [$]Z=0[$] outwards on both sides. First of all, I define how we calculate the coefficient of first hermite polynomial since this is the most important part, though quite simple. In this part, we add together the effect of first derivative outwards starting from Z=0. Here is how I do it. This procedure can slightly change depending upon how the grid is defined.
I want to mention that entire grid [$]w_n[$] as a function of [$] Z_n[$] is already given to us.
At Z=0, we do not need to define [$] c_{n1}=c_{01}[$] which is the coefficient of first hermite polynomial at Z=0.
at n=1, where [$]Z_n=Z_1=\Delta Z[$], we use the formula
[$]c_{11}=(\frac{dw}{dZ}(Z_0) \frac{\Delta Z}{2} + \frac{dw}{dZ}(Z_1) \frac{\Delta Z}{2} )/{\Delta Z}[$]
[$]=(\frac{1}{2} \frac{dw}{dZ}(Z_0) +\frac{1}{2} \frac{dw}{dZ}(Z_1) )[$]
Similarly at n=2 where [$]Z_n=Z_2[$] we calculate the coefficients as
[$]c_{21}=(c_{11} Z_1 + \frac{dw}{dZ}(Z_1) \frac{\Delta Z}{2} + \frac{dw}{dZ}(Z_2) \frac{\Delta Z}{2} )/{(Z_2 - Z_0)}[$]
where [$](Z_2 - Z_0)=Z_2[$] in above since [$]Z_0=0[$]
for the n+1 th point in positive Z direction, we will use the formula
[$]c_{(n+1)1}=(c_{n1} Z_n + \frac{dw}{dZ}(Z_n) \frac{\Delta Z}{2} + \frac{dw}{dZ}(Z_{(n+1)}) \frac{\Delta Z}{2} )/{(Z_{(n+1)} - Z_0)}[$]

In a very similar way, we can find coefficients of first order hermite polynomials in negative Z direction with negative indices. For example we could write the equation for coefficients of first hermite polynomial in negative Z part as

[$]c_{-(n+1)1}=(c_{-n1} Z_{-n} - \frac{dw}{dZ}(Z_{-n}) \frac{\Delta Z}{2} - \frac{dw}{dZ}(Z_{-(n+1)}) \frac{\Delta Z}{2} )/{(Z_{-(n+1)} - Z_0)}[$]

So we have calculated the coefficients with first order effect which require the knowledge of first derivatives across the grid. Now we turn to calculation of constant coefficient and the coefficients of the 2nd hermite polynomial. For this we consider the mid three points [$]w_1, w_0, w_{-1}[$] associated with [$]Z_1, Z_0, Z_{-1}[$]
We have
[$]w_0=w(Z_0)=c_0+c_{01} Z_0 +c_{02} ({Z_0}^2-1)=c_0 -c_{02}[$]
[$]w_1=w(Z_1)=c_0+c_{11} Z_1 +c_{12} ({Z_1}^2-1)[$]
[$]w_{-1}=w(Z_{-1})=c_0+c_{-11} Z_{-1} +c_{-12} ({Z_{-1}}^2-1)[$]

For the second hermite polynomial coefficents, we put together all the residual effect after the first order effect in terms of coefficients of second hermite polynomial as this would not require the calculation of second derivative something which can become hard to calculate analytically with precision.
We write the following three equations for three unknowns in the middle three grid points of w as

[$]c_{12}=((w_1 - w_0 ) - c_{11} Z_1 - c_{02})/({Z_1}^2-1)[$]
[$]c_{-12}=((w_{-1} - w_0 ) - c_{-11} Z_{-1} - c_{02})/({Z_{-1}}^2-1)[$]
and
[$]c_{02}=\frac{1}{2} c_{12} + \frac{1}{2} c_{-12}[$] since we expect enough linearity in coefficients
The above are three simple equations in three unknowns. And then we can find
[$]c_0=w_0+c_{02}[$]
Coefficients of second hermite polynomial associated with a general nth grid point would be calculated as
[$]c_{n2}=((w_n - w_0 ) + c_{n1} Z_n + c_{02})/({Z_n}^2-1)[$]
The above procedure will turn unstable close to Z=1 where we easily can interpolate the coefficients since second order hermite polynomial coefficients are independent of values of these coefficients at previous grid points in our set up.

After finding a general representation of a grid point in terms of first two hermite polynomials as
[$]w_n=c_0 + c_{n1} Z_n + c_{n2} ({Z_n}^2-1)[$]

we can find out the hermite representation of drift using a similar procedure and represent the drift as a function of hermite polynomials

[$]\mu_n(t)=d_0 + d_{n1} Z_n + d_{n2} ({Z_n}^2-1)[$]
where [$]\mu_n(t)[$] is the value of drift associated with the nth grid point where [$]Z_n[$] is the associated value of Z.
In what follows I have supposed that mean of the distribution w(t) can be represented as
[$]w_m=c_0 + c_{m1} Z_n + c_{m2} ({Z_m}^2-1)[$]
here [$]Z_m[$] is the value of Z associated with the mean of the distribution and [$]w_m[$] represents the mean of the distribution itself.

Then we can write our update equation as

[$] w(t+\Delta t)=\mu_w +d_0 + \sqrt{((c_{n1} Z_n-c_{m1} Z_m )^2+ (d_{n1} Z_n)^2 + (\Delta w)^2)}[$]
[$]+ \sqrt{(( c_{n2} (Z_n^2-1)-c_{m2} (Z_m^2-1) )^2 + ( d_{n2} (Z_n^2-1))^2)} [$]

where [$]\Delta w[$] represents the volatility effect which is quite close to linear as we have been using in our update equation earlier.
The above version of the equation might be slightly wrong and I have not thought very carefully about it yet. I wrote it like this since we are calculating variance from the mean and not from Z=0. Second hermite squared addition might possibly be calculated from zero as

[$] w(t+\Delta t)=\mu_w +d_0 + \sqrt{((c_{n1} Z_n-c_{m1} Z_m )^2+ (d_{n1} Z_n)^2 + (\Delta w)^2)}[$]
[$]+ \sqrt{(( c_{n2} (Z_n^2-1) )^2 + ( d_{n2} (Z_n^2-1))^2)} [$]

while carefully accounting for signs of each term in the equation as we have previously done in the older set up.
Also I think that instability at zero might be because at Z=0 and very close to zero, our previous set up assumes an approximation of the form
[$]w_0=w(Z_0)=c_0+c_{01} Z_0 =c_0[$]
and misses the contribution from second hermite polynomial and the right equation should be
[$]w_0=w(Z_0)=c_0+c_{01} Z_0 +c_{02} ({Z_0}^2-1)=c_0 -c_{02}[$]
which would alter the dynamics a slight bit close to zero also because [$]c_{02}[$] increases with time.

Another possibility could be that derivatives of drift are really not contributing in a squared fashion in the update but even then our new update equation would change and it would be given as below since second hermite contribution in our linearized lamperti/bessel form is extremely miniscule.

[$] w(t+\Delta t)=\mu_w +\mu(t) + \sqrt{((c_{n1} Z_n-c_{m1} Z_m )^2+ (\Delta w)^2)}[$]
[$]+ c_{n2} (Z_n^2-1) [$]

In next few days, I will be experimenting with above form of solution and report the analysis and post new program.

Amin · March 11th, 2020, 8:41 pm

There were minor errors in notation in previous post that I corrected. Here is another change I made but first I copy a small part of previous post

For the second hermite polynomial coefficents, we put together all the residual effect after the first order effect in terms of coefficients of second hermite polynomial as this would not require the calculation of second derivative something which can become hard to calculate analytically with precision.
We write the following three equations for three unknowns in the middle three grid points of w as

[$]c_{12}=((w_1 - w_0 ) - c_{11} Z_1 - c_{02})/({Z_1}^2-1)[$]
[$]c_{-12}=((w_{-1} - w_0 ) - c_{-11} Z_{-1} - c_{02})/({Z_{-1}}^2-1)[$]
and
[$]c_{02}=\frac{1}{2} c_{12} + \frac{1}{2} c_{-12}[$] since we expect enough linearity in coefficients
The above are three simple equations in three unknowns. And then we can find
[$]c_0=w_0+c_{02}[$]
Coefficients of second hermite polynomial associated with a general nth grid point would be calculated as
[$]c_{n2}=((w_n - w_0 ) + c_{n1} Z_n + c_{02})/({Z_n}^2-1)[$]

In the above, instead of
[$]c_{02}=\frac{1}{2} c_{12} + \frac{1}{2} c_{-12}[$]
I used
[$]c_{02}=.5 \frac{\partial^2 w}{\partial Z^2} |^{Z=0} [$]

I am posting the program for this decomposition into hermite polynomials here. But with disclaimer that I have not extensively tested the program so it is only for friends who are doing research on this.

function [C0,C1,C2,w0,C20,wMean,ZMean,nnMean] = CalculateHermitesOrderTwo(w,Z,Prob,wnStart,Nn,dNn,NnMidh,NnMidl)

%Mean is calculated and returned but it has nothing to do with calculation
%of hermite coefficients. It is for future use.
%--------------------------------------------------------------------
wMean=sum(w(wnStart:Nn).*Prob(wnStart:Nn));  %Calculate the mean. This is in general 

ZMean=(w(NnMidh)+w(NnMidl))/2.0;
nnMean=NnMidl;
for nn=wnStart:Nn-1
    if( (w(nn+1)>wMean) && (w(nn)<=wMean))
        ZMean=Z(nn)+(wMean-w(nn))/(w(nn+1)-w(nn))*(Z(nn+1)-Z(nn));
        nnMean=nn;
    end
end
%-------------------------------------------------------------------------
%Below hermite coefficients calculation starts. InterpolateOrderN is a
%program that interpolates the values of w0 at Z=0 since in our set up grid
%points are located at Z(NnMidl)=-.1 and Z(NnMidh)=.1 and there are no grid
%points at Z=0.

Z0=0;
w0= InterpolateOrderN(4,Z0,Z(NnMidh),Z(NnMidh+1),Z(NnMidl),Z(NnMidl-1),0,w(NnMidh),w(NnMidh+1),w(NnMidl),w(NnMidl-1),0);
dwdZ0=(w(NnMidh)-w(NnMidl))./dNn;
d2wdZ20 = (-1*w(NnMidl-1)+81*w(NnMidl)-160*w0+81*w(NnMidh)-1*w(NnMidh+1))/(72*(.5*dNn)^2);

%Calculate first hermite polynomial coefficients C1 below.
dwdZ(wnStart)=(-3*w(wnStart)+4*w(wnStart+1)-1*w(wnStart+2))/(2*dNn);
dwdZ(wnStart+1:Nn-1)=(-1*w(wnStart:Nn-2)+1*w(wnStart+2:Nn))/(2*dNn);
dwdZ(Nn)=(3*w(Nn)-4*w(Nn-1)+1*w(Nn-2))/(2*dNn);

C1(wnStart:Nn)=0;
C1(NnMidh)=.5*dwdZ0+.5*dwdZ(NnMidh);
for nn=NnMidh+1:Nn
        C1(nn)=(C1(nn-1).*Z(nn-1)+(.5*dwdZ(nn-1)+.5*dwdZ(nn))*dNn)./Z(nn);
end
C1(NnMidl)=.5*dwdZ0+.5*dwdZ(NnMidl);
for nn=NnMidl-1:-1:wnStart
        C1(nn)=(C1(nn+1).*Z(nn+1)-(.5*dwdZ(nn+1)+.5*dwdZ(nn))*dNn)./Z(nn);
end

%Calculate second hermite coefficients below. Here C20 are coefficients of
%second hermite polynomial at Z=0 
C20=.5*d2wdZ20;
C0=w0+C20;

for nn=NnMidh:Nn
    C2(nn)=((w(nn)-w0)-C1(nn).*Z(nn)-C20)./(Z(nn).^2-1);
end
for nn=NnMidl:-1:wnStart
    C2(nn)=((w(nn)-w0)-C1(nn).*Z(nn)-C20)./(Z(nn).^2-1);
end

w2(wnStart:Nn)=C0+C1(wnStart:Nn).*Z(wnStart:Nn)+C2(wnStart:Nn).*(Z(wnStart:Nn).^2-1);

%Below we plot a graphical comparison of the original SDE variable w and
%the variable w1 reproduced from the coefficients of the hermite
%polynomials
 plot(Z(wnStart:Nn),w(wnStart:Nn),'g',Z(wnStart:Nn),w2(wnStart:Nn),'r')
%Sigma00

C1
C2
C0
 str=input('Look at the graphical comparison of w and w1 as a function of standard gaussian Z');
end

Here is a dependency function that you will need for the above program.

function [y0] = InterpolateOrderN(N,x0,x1,x2,x3,x4,x5,y1,y2,y3,y4,y5)


if(N==5)
y0=(x0-x2)*(x0-x3)*(x0-x4)*(x0-x5)/((x1-x2).*(x1-x3).*(x1-x4).*(x1-x5))*y1+ ...
   (x0-x1)*(x0-x3)*(x0-x4)*(x0-x5)/((x2-x1).*(x2-x3).*(x2-x4).*(x2-x5))*y2+ ...
   (x0-x1)*(x0-x2)*(x0-x4)*(x0-x5)/((x3-x1).*(x3-x2).*(x3-x4).*(x3-x5))*y3+ ...
   (x0-x1)*(x0-x2)*(x0-x3)*(x0-x5)/((x4-x1).*(x4-x2).*(x4-x3).*(x4-x5))*y4+ ...
   (x0-x1)*(x0-x2)*(x0-x3)*(x0-x4)/((x5-x1).*(x5-x2).*(x5-x3).*(x5-x4))*y5;
end
if(N==4)
y0=(x0-x2)*(x0-x3)*(x0-x4)/((x1-x2).*(x1-x3).*(x1-x4))*y1+ ...
   (x0-x1)*(x0-x3)*(x0-x4)/((x2-x1).*(x2-x3).*(x2-x4))*y2+ ...
   (x0-x1)*(x0-x2)*(x0-x4)/((x3-x1).*(x3-x2).*(x3-x4))*y3+ ...
   (x0-x1)*(x0-x2)*(x0-x3)/((x4-x1).*(x4-x2).*(x4-x3))*y4;
end
if(N==3)
y0=(x0-x2)*(x0-x3)/((x1-x2).*(x1-x3))*y1+ ...
   (x0-x1)*(x0-x3)/((x2-x1).*(x2-x3))*y2+ ...
   (x0-x1)*(x0-x2)/((x3-x1).*(x3-x2))*y3;
end
if(N==2)
y0=(x0-x2)/((x1-x2))*y1+ ...
   (x0-x1)/((x2-x1))*y2;
end

%y0=(x0-x3)/(x2-x3)*y2+(x0-x2)/(x3-x2)*y3;

end

Amin · March 12th, 2020, 8:47 pm

Here is a new program with different hermite effects separated from each other. I want to extend my hermite representation to third order from the current second order and I am very sure that the new program will work extremely well after that. The program has a simplified structure and first hermite is added in a squared fashion to first hermite and 2nd hermite is added in a squared fashion to 2nd hermite as these effects have been successfully separated (there is still a fudge term with a fudge constant mentioned in the program.)
Here is the simplified new program.

function [TrueMean,ItoHermiteMean,MCMean] = ItoHermiteMethodWilmott08ANewGood0440Q2(x0,theta,kappa,gamma,sigma0,T)

%Copyright Ahsan Amin. Infiniti derivatives Technologies.
%Please fell free to connect on linkedin: linkedin.com/in/ahsan-amin-0a53334 
%or skype ahsan.amin2999
%In this program, I am simulating the SDE given as
%dy(t)=mu1 x(t)^beta1 dt + mu2 x(t)^beta2 dt +sigma x(t)^gamma dz(t)

%I have not directly simulated the SDE but simulated the transformed 
%Besse1l process version of the SDE and then changed coordinates to retreive
%the SDE in original coo
%rdinates.
%The present program will analytically evolve only the Bessel Process version of the
%SDE in transformed coordinates.
%In this version, the mean correction has been disabled. If you are
%simulating standard mean-reverting SV type SDEs, please enable the mean 
%correction by uncommenting the appropriate line in the body of code below.

dt=.125/16;   % Simulation time interval.%Fodiffusions close to zero
             %decrease dt for accuracy.
%T=5;
Tt=128*T;%16*16;     % Number of simulation levels. Terminal time= Tt*dt; //.125/32*32*16=2 year; 
OrderA=2;  %Keep at two.
OrderM=2;  %Keep at two.
%dtM=.125/8;%Monte carlo time interval size dtM.
%TtM=8*8;%Monte carlo number of simulation intervals.
dtM=dt*2;
TtM=Tt/2;


dNn=.2/1;   % Normal density subdivisions width. would change with number of subdivisions
Nn=48;  % No of normal density subdivisions
NnMidl=25;%One half density Subdivision left from mid of normal density(low)
NnMidh=26;%One half density subdivision right from the mid of normal density(high)
NnMid=4.0;

%theta=mu1/(-mu2);
%kappa=-mu2;

%x0=.25;   % starting value of SDE
beta1=0.0;
beta2=1.0;   % Second drift term power.
%gamma=.95;%50;   % volatility power.                                                                                                                                                                                                                                                                     
%kappa=.5;   %mean reversion parameter.
%theta=.075;%mean reversion target

%you can specify any general mu1 and mu2 and beta1 and beta2.
%The present program is not perfect but quite reasonably good.
%if there is only one term in drift with beta=0, there will be minor
%errors. This case is yet not covered.

mu1=+1*theta*kappa;   %first drift coefficient.
mu2=-1*kappa;    % Second drift coefficient.
%sigma0=1.0;%Volatility value
alpha=1;% x^alpha is being expanded. This is currently for monte carlo only.
alpha1=1-gamma;
                
w(1:Nn)=x0^(1-gamma)/(1-gamma);
q(1:Nn)=x0^(1-gamma)/(1-gamma);
Z(1:Nn)=(((1:Nn)-5.5)*dNn-NnMid);

Z
str=input('Look at Z');
ZProb(1)=normcdf(.5*Z(1)+.5*Z(2),0,1)-normcdf(.5*Z(1)+.5*Z(2)-dNn,0,1);
ZProb(Nn)=normcdf(.5*Z(Nn)+.5*Z(Nn-1)+dNn,0,1)-normcdf(.5*Z(Nn)+.5*Z(Nn-1),0,1);

for nn=2:Nn-1
    ZProb(nn)=normcdf(.5*Z(nn)+.5*Z(nn+1),0,1)-normcdf(.5*Z(nn)+.5*Z(nn-1),0,1);
    %Above calculate probability mass in each probability subdivision.
end

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%


wnStart=1;%
d2wdZ2(1:Nn)=0;
dwdZ(1:Nn)=0;
yy(wnStart:Nn)=((1-gamma)*w(wnStart:Nn)).^(1/(1-gamma));
tic

for tt=1:Tt
  
     yy(wnStart:Nn)=((1-gamma)*w(wnStart:Nn)).^(1/(1-gamma));
    [wMu0dt,dwMu0dtdw,d2wMu0dtdw2,c1,c2] = CalculateDriftAndDerivatives(w,wnStart,Nn,mu1,mu2,beta1,beta2,sigma0,gamma,dt);

    dw(wnStart:Nn)=c1(wnStart:Nn).*Z(wnStart:Nn) ;% ...
    dw2(wnStart:Nn)=dw(wnStart:Nn).^2;
    d2w(wnStart:Nn)=c2(wnStart:Nn).*(Z(wnStart:Nn).^2-1);
    
    w(isnan(w)==1)=0;
    
    wMu0dt(isnan(wMu0dt)==1)=0;  
    wMeanPrev=sum(ZProb(wnStart:Nn).*w(wnStart:Nn));%Calculate the mean.
   
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%



[wMean,ZMean,Sigma00,Sigma11,Sigma22,Sigma33,Sigma1,Sigma2,Sigma3,Sigma0,w1New,w2New,nnMean] = CalculateHermiteRepresentationOfDensityOrderThree(w,Z,ZProb,wnStart,Nn,dNn,NnMidh,NnMidl);

d2wdZ2(wnStart:Nn)=d2wdZ2(wnStart:Nn)+ d2wMu0dtdw2(wnStart:Nn).*dwdZ(wnStart:Nn).^2+dwMu0dtdw(wnStart:Nn).*d2wdZ2(wnStart:Nn);%+ ...

dwdZ(wnStart:Nn)=(Sigma00)+Sigma1(wnStart:Nn).*2.*Z(wnStart:Nn);

%dwdZN(wnStart+1:Nn-1)=(w(wnStart+2:Nn)-w(wnStart:Nn-2))/(2*dNn);
%dwdZN(wnStart)= InterpolateOrderN(4,Z(wnStart),Z(wnStart+1),Z(wnStart+2),Z(wnStart+3),Z(wnStart+4),0,dwdZN(wnStart+1),dwdZN(wnStart+2),dwdZN(wnStart+3),dwdZN(wnStart+4),0);
    

H2(wnStart:Nn)=(Z(wnStart:Nn).^2-1);
      
CTerm(wnStart:Nn)=(d2wMu0dtdw2(wnStart:Nn).*dwdZ(wnStart:Nn).^2 - ...
          dwMu0dtdw(wnStart:Nn).*d2wdZ2(wnStart:Nn)); 
      
Correction1(wnStart:Nn)=-1/sqrt(2)/(22/7)^2.*CTerm(wnStart:Nn).*H2(wnStart:Nn);      
      
if(tt==1)
w(wnStart:Nn)=wMeanPrev+wMu0dt(wnStart:Nn)+ ...
          sign(w(wnStart:Nn)-wMeanPrev+dw(wnStart:Nn)).* ...
          sqrt(abs(sign(w(wnStart:Nn)-wMeanPrev).*(w(wnStart:Nn)-wMeanPrev).^2+ ...
          sign(dw(wnStart:Nn)).*dw2(wnStart:Nn)));
end
if(tt>1)
    
    
     [C0,C1,C2,w0,C20,wMean,ZMean,nnMean,Cm1,Cm2] = CalculateHermitesOrderTwo(w,Z,ZProb,wnStart,Nn,dNn,NnMidh,NnMidl);
     %Below is hermite decomposition of w
     %w(wnStart:Nn)=C0+C1(wnStart:Nn).*Z(wnStart:Nn)+C2(wnStart:Nn).*(Z(wnStart:Nn).^2-1);
     %Below is hermite decomposition of wMean with associated ZMean.
     %wMean=C0+Cm1.*ZMean+Cm2.*(ZMean^2-1);
     
     [mC0,mC1,mC2,mw0,mC20] = CalculateHermitesOrderTwoWithoutMean(wMu0dt,Z,ZProb,wnStart,Nn,dNn,NnMidh,NnMidl);
     %Below is hermite decomposition of drift   
     %wMu0dt(wnStart:Nn)=mC0+mC1(wnStart:Nn).*Z(wnStart:Nn)+mC2(wnStart:Nn).*(Z(wnStart:Nn).^2-1);
    
     CFudge=.5/pi^2; %Fudge constant
 
 w(wnStart:Nn)=wMeanPrev+wMu0dt(wnStart:Nn)+ ...
         sign(C1(wnStart:Nn).*Z(wnStart:Nn)-Cm1.*ZMean+dw(wnStart:Nn)).* ...
         sqrt(abs(sign(C1(wnStart:Nn).*Z(wnStart:Nn)-Cm1.*ZMean).*(C1(wnStart:Nn).*Z(wnStart:Nn)-Cm1.*ZMean).^2+ ...
         sign(dw(wnStart:Nn)).*dw2(wnStart:Nn)))+ ...
         sign(C2(wnStart:Nn).*(Z(wnStart:Nn).^2-1)-Cm2.*(ZMean.^2-1)+d2w(wnStart:Nn)+sqrt(CFudge)*mC2(wnStart:Nn).*(Z(wnStart:Nn).^2-1)).* ...
         sqrt(abs(sign(C2(wnStart:Nn).*(Z(wnStart:Nn).^2-1)-Cm2.*(ZMean.^2-1)).*(C2(wnStart:Nn).*(Z(wnStart:Nn).^2-1)-Cm2.*(ZMean.^2-1)).^2- ...
         sign(d2w(wnStart:Nn)).*d2w(wnStart:Nn).^2 + ...
         CFudge*sign(mC2(wnStart:Nn).*(Z(wnStart:Nn).^2-1)).*(mC2(wnStart:Nn).*(Z(wnStart:Nn).^2-1)).^2))+...
         Correction1(wnStart:Nn);
        

end         
  
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%    
%Below are instability correction equations. There is instability in the 
% SDE representation equation roughly between -.5 SD to +.5 SD in normal terms.
%Instability correction Algorithm takes two ends that are here given as
%lower end=w(NnMidl-2) and upper end=w(NnMidh+3).
%In the block below the interpolation starts from lower end.
%1st equation is
%wll(NnMidl-1)=w(NnMidl-2)+(D1wl+CTheta*D2wl*dNn/1+.5*DeltaD1w)*dNn;
%Second equation is: wll(NnMidl)=wll(NnMidl-1)+(D1wl+CTheta*D2wl*dNn+1.5*DeltaD1w)*dNn;
%And so on.
%There is a sixth equation that is not written below(since it takes us to 
%the upper interpolation point which is already known) which is 
%wll(NnMidh+3)=wll(NnMidh+2)+(D1wl+CTheta*D2wl*dNn+5.5*DeltaD1w)*dNn;
%Sum of these six equations add up to the distance between upper and lower
%interpolation points.
%In the sum there are six instances of first derivative which is already 
%calculated at the boundary, six instances of second derivative which is also
%already calculated at lower boundary and 18.0 instances of DeltaD1w which is 
%unknown. This equation calculates the value of one unit of DeltaD1w addition
%DeltaD1w=(w(NnMidh+3)-w(NnMidl-2)-6*D1wl*dNn-6.0*CTheta*D2wl*dNn.^2.0/1.0)/18.0/dNn;
% I hope to find a better analytical method to tackle this issue. Friends
% are welcome to play with slightly changed related alternatives to the 
%above interpolation.
    CTheta=1.0;
    D1wl=(11/6*w(NnMidl-2)-3*w(NnMidl-3)+3/2*w(NnMidl-4)-1/3*w(NnMidl-5))/dNn;
    D2wl=(35/12*w(NnMidl-2) -26/3*w(NnMidl-3)+ 19/2*w(NnMidl-4)-14/3*w(NnMidl-5)+11/12*w(NnMidl-6))/dNn.^2;
    DeltaD1w=(w(NnMidh+3)-w(NnMidl-2)-6*D1wl*dNn-6.0*CTheta*D2wl*dNn.^2.0/1.0)/18.0/dNn;
    
    wll(NnMidl-1)=w(NnMidl-2)+  (D1wl+CTheta*D2wl*dNn+.5*DeltaD1w)*dNn;
    wll(NnMidl)=wll(NnMidl-1)+  (D1wl+CTheta*D2wl*dNn+1.5*DeltaD1w)*dNn;
    wll(NnMidh)=wll(NnMidl)+    (D1wl+CTheta*D2wl*dNn+2.5*DeltaD1w)*dNn;
    wll(NnMidh+1)=wll(NnMidh)+  (D1wl+CTheta*D2wl*dNn+3.5*DeltaD1w)*dNn;
    wll(NnMidh+2)=wll(NnMidh+1)+(D1wl+CTheta*D2wl*dNn+4.5*DeltaD1w)*dNn;
    %wll(NnMidh+3)=wll(NnMidh+2)+(D1wl+CTheta*D2wl*dNn+5.5*DeltaD1w)*dNn;
%The last equation is not used since point w(NnMidh+3)is given but it is used in 
%calculation of distance between two interpolation points. This distance is
%used to determine the value of DeltaD1w.

    D1wh=(-11/6*w(NnMidh+2)+3*w(NnMidh+3)-3/2*w(NnMidh+4)+1/3*w(NnMidh+5))/dNn;
    D2wh=(35/12*w(NnMidh+2) -26/3*w(NnMidh+3)+ 19/2*w(NnMidh+4)-14/3*w(NnMidh+5)+11/12*w(NnMidh+6))/dNn.^2;
    DeltaD1w=(w(NnMidh+3)-w(NnMidl-2)-6*D1wh*dNn-6.0*CTheta*D2wh*dNn.^2.0/1.0)/18.0/dNn;
%     DeltaD1w=(w(NnMidh+2)-w(NnMidl-1)-4*D1wh*dNn-16.0*D2wh*dNn.^2.0/1.0)/6.0/dNn;


    whh(NnMidh+2)=w(NnMidh+3)-(D1wh+CTheta*D2wh*dNn+.5*DeltaD1w)*dNn;
    whh(NnMidh+1)=w(NnMidh+2)-(D1wh+CTheta*D2wh*dNn+1.5* DeltaD1w)*dNn;
    whh(NnMidh)=whh(NnMidh+1)-(D1wh+CTheta*D2wh*dNn+2.5* DeltaD1w)*dNn;
    whh(NnMidl)=whh(NnMidh)-(D1wh+CTheta*D2wh*dNn+3.5* DeltaD1w)*dNn;
    whh(NnMidl-1)=whh(NnMidl)-(D1wh+CTheta*D2wh*dNn+4.5* DeltaD1w)*dNn;

    %whh(NnMidl-2)=whh(NnMidl-1)-(D1wh+CTheta*D2wh*dNn/1+5.5* DeltaD1w)*dNn;

% 
     w(NnMidh+2)=.5*whh(NnMidh+2)+.5*wll(NnMidh+2);
     w(NnMidh+1)=.5*whh(NnMidh+1)+.5*wll(NnMidh+1);
     w(NnMidh)=.5*whh(NnMidh)+.5*wll(NnMidh);
     w(NnMidl)=.5*whh(NnMidl)+.5*wll(NnMidl);
     w(NnMidl-1)=.5*whh(NnMidl-1)+.5*wll(NnMidl-1);



             
        w1(1:Nn-1)=w(1:Nn-1);
        w2(1:Nn-1)=w(2:Nn);
        w(w1(:)>w2(:))=0;%Be careful;might not universally hold;
     
        w(w<0)=0.0;
        for nn=1:Nn
            if(w(nn)<=0)
                wnStart=nn+1;
           end
        end
        
        %%1st order mean correction. We know the mean in original
        %%coordinates so I shift the density into original coordinates,
        %%apply the mean correction and then transform again into Lamperti
        %%coordinates. Algebra solves two equations in two unknowns.
        %%Equations are Sum_0^N{(Y_w(wnStart:Nn)-Y0)*W0}=1 and the second
        %%equation is Sum_0^N{Y_w(wnStart:Nn).*(Y_w(wnStart:Nn)-Y0)*W0}=u0
        %%Two unknows are Y0 and W0. u0 is known mean.
        tt;
        u0=theta+(x0-theta)*exp(-kappa*(tt*dt)); %analytic mean of the density
        
        %If you are not using stochastic volatility, replace above with
        %true mean otherwise results would become garbage. 

        Y_w(wnStart:Nn) = ((1-gamma)*w(wnStart:Nn)).^(1/(1-gamma));
    
        Yn2Pn=sum(Y_w(wnStart:Nn).*Y_w(wnStart:Nn).*ZProb(wnStart:Nn));
        YnPn=sum(Y_w(wnStart:Nn).*ZProb(wnStart:Nn));
        Pn=1.0;%%Sum(ZProb(1:Nn))
        Y0=(Yn2Pn-u0*YnPn)/(YnPn-u0*Pn);
    
        Y0Pn=Y0*Pn;
        W0=1/(YnPn-Y0Pn);
        YCorrected(wnStart:Nn)=Y_w(wnStart:Nn).*(Y_w(wnStart:Nn)-Y0)*W0;
        wCorrected(wnStart:Nn)=(YCorrected(wnStart:Nn).^(1-gamma))./(1-gamma);
   %     w(wnStart:Nn)=wCorrected(wnStart:Nn);
        
   %I have disabled the mean correction. The above mean correction is only valid for 
   %standard mean reverting stochastic volatility type SDEs. To enable mean
   %correction please uncomment the above last line in the block.
        
        
end

 y_w(1:Nn)=0;
 y_w(wnStart:Nn) = ((1-gamma)*w(wnStart:Nn)).^(1/(1-gamma));
 Dfy_w(wnStart:Nn)=0;
 Dfw(wnStart:Nn)=0;
for nn=wnStart+1:Nn-1
    Dfy_w(nn) = (y_w(nn + 1) - y_w(nn - 1))/(Z(nn + 1) - Z(nn - 1));
    Dfw(nn) = (w(nn + 1) - w(nn - 1))/(Z(nn + 1) - Z(nn - 1));
    %Change of variable derivative for densities
end
fy_w(1:Nn)=0;
fw(1:Nn)=0;
for nn = wnStart:Nn-1
    fy_w(nn) = (normpdf(Z(nn),0, 1))/abs(Dfy_w(nn));%Origianl coordinates density
    fw(nn) = (normpdf(Z(nn),0, 1))/abs(Dfw(nn));
end

toc
%str=input('Analytic Values calculated; Press a key to start monte carlo');


sigma11(1:OrderA+1)=0;
mu11(1:OrderA+1)=0;
mu22(1:OrderA+1)=0;
sigma22(1:OrderA+1)=0;
% index 1 correponds to zero level since matlab indexing starts at one. 
sigma11(1)=1;
mu11(1)=1;
mu22(1)=1;
sigma22(1)=1;

for k=1:(OrderA+1)
    if sigma0~=0
        sigma11(k)=sigma0^(k-1);
    end
    if mu1 ~= 0
        mu11(k)=mu1^(k-1);
    end
    if mu2 ~= 0
        mu22(k)=mu2^(k-1);
    end
    if sigma0~=0
        sigma22(k)=sigma0^(2*(k-1));
    end
end
Ft(1:TtM+1,1:(OrderA+1),1:(OrderA+1),1:(OrderA+1),1:(OrderA+1))=0; %General time powers on hermite polynomials
Fp(1:(OrderA+1),1:(OrderA+1),1:(OrderA+1),1:(OrderA+1))=0;%General x powers on coefficients of hermite polynomials.
Fp1(1:(OrderA+1),1:(OrderA+1),1:(OrderA+1),1:(OrderA+1))=0;%General x powers for bessel transformed coordinates.
%Pre-compute the time and power exponent values in small multi-dimensional arrays
for k = 0 : (OrderA+1)
    for m = 0:k
        l4 = k - m + 1;
        for n = 0 : m
            l3 = m - n + 1;
            for j = 0:n
                l2 = n - j + 1;
                l1 = j + 1;
                Ft(l1,l2,l3,l4) = dtM^((l1-1) + (l2-1) + (l3-1) + .5* (l4-1));
                Fp(l1,l2,l3,l4) = (alpha + (l1-1) * beta1 + (l2-1) * beta2 + (l3-1) * 2* gamma + (l4-1) * gamma ...
                    - (l1-1) - (l2-1) - 2* (l3-1) - (l4-1));
                Fp1(l1,l2,l3,l4) = (alpha1 + (l1-1) * beta1 + (l2-1) * beta2 + (l3-1) * 2* gamma + (l4-1) * gamma ...
                    - (l1-1) - (l2-1) - 2* (l3-1) - (l4-1));

            end
        end
    end
end
%Below call the program that calculates the Ito-Taylor coeffs.
YCoeff = ItoTaylorCoeffsNew(alpha,beta1,beta2,gamma);





%Yq=ItoTaylorCoeffsNew(alpha1,beta1,beta2,gamma);
%YqCoeff=Yq/(1-gamma);%These coordinates are divided by (1-gamma)
 %Calculate the  density with monte carlo below.
 %rng(85109634, 'twister')
 rng(29079137, 'twister')
 paths=100000;
 YY(1:paths)=x0;  %Original process monte carlo.
% YY2dt(1:paths)=0.0;
 %YY4dt(1:paths)=0.0;
 X(1:paths)=x0^(1-gamma)/(1-gamma);
 Random1(1:paths)=0;
 YYMean(1:TtM)=0;
 for tt=1:TtM
 %    t1=tt*dtM;
 %    t0=(tt-1)*dtM;
     Random1=randn(size(Random1));
     HermiteP1(1,1:paths)=1;
     HermiteP1(2,1:paths)=Random1(1:paths);
     HermiteP1(3,1:paths)=Random1(1:paths).^2-1;
     HermiteP1(4,1:paths)=Random1(1:paths).^3-3*Random1(1:paths);
     HermiteP1(5,1:paths)=Random1(1:paths).^4-6*Random1(1:paths).^2+3;
     YY0=YY;
     YY1=YY;
  %   YYPrev=YY;
  %   XX0=X;
     
     for k = 0 : OrderM
         for m = 0 : k
             l4 = k - m + 1;
             for n = 0 : m
                 l3 = m - n + 1;
                 for j = 0 : n
                     l2 = n - j + 1;
                     l1 = j + 1;
                     if(l4<=5)   
                         
  %                    X(1:paths)=X(1:paths) + YqCoeff(l1,l2,l3,l4) * ...
  %                   mu11(l1)*mu22(l2)*sigma22(l3)*sigma11(l4)* ...
  %                   ((1-gamma)*XX0(1:paths)).^(Fp1(l1,l2,l3,l4)/(1-gamma)) .* ...
  %                   HermiteP1(l4,1:paths) * Ft(l1,l2,l3,l4);    
                 
                     YY(1:paths)=YY(1:paths) + YCoeff(l1,l2,l3,l4) * ...
                     mu11(l1)*mu22(l2)*sigma22(l3)*sigma11(l4)* ...
                     YY0(1:paths).^Fp(l1,l2,l3,l4) .* HermiteP1(l4,1:paths) * Ft(l1,l2,l3,l4);
%                     YY(YY>0)=YY1(YY>0);
%                     YY(YY<0)=0;
                 %Above is original coordinates monte carlo equation.
                     end
                 end
             end
         end
     end
     %YY(YY<0)=0;
     YYMean(tt)=YYMean(tt)+sum(YY(1:paths))/paths;
 end
 YY(YY<0)=0;
 disp('Original process average from monte carlo');
 MCMean=sum(YY(:))/paths %origianl coordinates monte carlo average.
 disp('Original process average from our simulation');
 ItoHermiteMean=sum(y_w(wnStart+1:Nn-1).*ZProb(wnStart+1:Nn-1)) %Original process average from coordinates 
 
 disp('true Mean only applicble to standard SV mean reverting type models otherwise disregard');
 TrueMean=theta+(x0-theta)*exp(-kappa*dt*Tt)%Mean reverting SDE original variable true average
 
% T11Mean=sum(T11(wnStart:Nn).*ZProb(wnStart:Nn));
% T21Mean=sum(T21(wnStart:Nn).*ZProb(wnStart:Nn));
% T31Mean=sum(T31(wnStart:Nn).*ZProb(wnStart:Nn));
 
 MaxCutOff=30;
 NoOfBins=round(500*gamma^2*1*sigma0);
 %[YDensity,IndexOutY,IndexMaxY] = MakeDensityFromSimulation_Infiniti(YY,paths,BinSize,MaxCutOff);
 [YDensity,IndexOutY,IndexMaxY] = MakeDensityFromSimulation_Infiniti_NEW(YY,paths,NoOfBins,MaxCutOff );
 plot(y_w(wnStart+1:Nn-1),fy_w(wnStart+1:Nn-1),'r',IndexOutY(1:IndexMaxY),YDensity(1:IndexMaxY),'g');
 %plot(y_w(wnStart+1:Nn-1),fy_w(wnStart+1:Nn-1),'r',IndexOutY(1:IndexMaxY),YDensity(1:IndexMaxY),'g',Z(wnStart+1:Nn-1),fy_w(wnStart+1:Nn-1),'b');
 title('Graphical Comparison of Ito-Hermite Method and Monte Carlo Method Density')
 legend({'Ito-Hermite Density','Monte Carlo Density'},'Location','northeast')
 
 str=input('red line is density of SDE from Ito-Hermite method, green is monte carlo.');
 

end

Here are several dependency programs. First function.

function [wMu0dt,dwMu0dtdw,d2wMu0dtdw2,c1,c2] = CalculateDriftAndDerivatives(w,wnStart,Nn,mu1,mu2,beta1,beta2,sigma0,gamma,dt)


 dt2=dt^2/2;
 dt3=dt^3/6.0;
 dtsqrt=sqrt(dt);
 dtonehalf=dt.^1.5;
    yy(wnStart:Nn)=((1-gamma)*w(wnStart:Nn)).^(1/(1-gamma));

    theta1=mu1;
    theta2=mu2;
    theta3=.5*sigma0^2*(-gamma);
    omega1=beta1-gamma;
    omega2=beta2-gamma;
    omega3=gamma-1;
    
    GG(wnStart:Nn)=theta1*yy(wnStart:Nn).^omega1+theta2*yy(wnStart:Nn).^omega2+theta3*yy(wnStart:Nn).^omega3;
    
    dGGdw(wnStart:Nn)=omega1* theta1 *((1-gamma)* w(wnStart:Nn)).^(-1+omega1/(1-gamma))+ ...
        omega2* theta2* ((1-gamma)* w(wnStart:Nn)).^(-1+omega2/(1-gamma))+ ...
        omega3* theta3* ((1-gamma)* w(wnStart:Nn)).^(-1+omega3/(1-gamma));
    
    
    d2GGdw2(wnStart:Nn)=(1 - gamma)* omega1* (-1 + omega1/(1 - gamma))* theta1* ((1 - gamma)* ...
        w(wnStart:Nn)).^(-2 + omega1/(1 - gamma)) + ...
        (1 - gamma)* omega2* (-1 + omega2/(1 - gamma))* theta2* ((1 - gamma)* w(wnStart:Nn)).^(-2 + omega2/(1 - gamma)) + ...
        (1 - gamma)* omega3* (-1 + omega3/(1 - gamma))* theta3* ((1 - gamma)* w(wnStart:Nn)).^(-2 + omega3/(1 - gamma));
    
     d3GGdw3(wnStart:Nn)=(1-gamma)^2* omega1* (-2+omega1/(1-gamma))* (-1+omega1/(1-gamma))* theta1* ...
         ((1-gamma)* w(wnStart:Nn)).^(-3+omega1/(1-gamma))+ ...
         (1-gamma)^2* omega2* (-2+omega2/(1-gamma))* (-1+omega2/(1-gamma))* theta2* ((1-gamma)* w(wnStart:Nn)).^(-3+omega2/(1-gamma))+ ...
         (1-gamma)^2* omega3* (-2+omega3/(1-gamma))* (-1+omega3/(1-gamma))* theta3* ((1-gamma)* w(wnStart:Nn)).^(-3+omega3/(1-gamma));
    
    
    dGG(wnStart:Nn)=omega1*theta1*yy(wnStart:Nn).^(omega1-1)+omega2*theta2*yy(wnStart:Nn).^(omega2-1) + ...
        +omega3*theta3*yy(wnStart:Nn).^(omega3-1);
    
    ddGGdw(wnStart:Nn)=(-1+omega1)* omega1* theta1* ((1-gamma)* w(wnStart:Nn)).^(-1+(-1+omega1)/(1-gamma))+ ...
        (-1+omega2)* omega2* theta2* ((1-gamma)* w(wnStart:Nn)).^(-1+(-1+omega2)/(1-gamma))+ ...
        (-1+omega3)* omega3* theta3* ((1-gamma)* w(wnStart:Nn)).^(-1+(-1+omega3)/(1-gamma));
    
     d2dGGdw2(wnStart:Nn)=(1-gamma)* (-1+(-1+omega1)/(1-gamma))* (-1+omega1)* omega1 * ...
         theta1* ((1-gamma)* w(wnStart:Nn)).^(-2+(-1+omega1)/(1-gamma))+ ...
         (1-gamma)* (-1+(-1+omega2)/(1-gamma))* (-1+omega2)* omega2 *theta2* ((1-gamma)* w(wnStart:Nn)).^(-2+(-1+omega2)/(1-gamma))+ ...
         (1-gamma)* (-1+(-1+omega3)/(1-gamma))* (-1+omega3)* omega3 *theta3* ((1-gamma)* w(wnStart:Nn)).^(-2+(-1+omega3)/(1-gamma));
     
     
     d3dGGdw3(wnStart:Nn)=(1 - gamma)^2* (-2 + (-1 + omega1)/(1 - gamma))* (-1 + (-1 + omega1)/(1 - gamma))* ...
         (-1 + omega1)* omega1* theta1* ((1 - gamma)* w(wnStart:Nn)).^(-3 + (-1 + omega1)/(1 - gamma)) + ...
         (1 - gamma)^2 *(-2 + (-1 + omega2)/(1 - gamma))* (-1 + (-1 + omega2)/(1 - gamma))* (-1 + omega2)* ...
         omega2* theta2* ((1 - gamma)* w(wnStart:Nn)).^(-3 + (-1 + omega2)/(1 - gamma)) + ...
         (1 - gamma)^2* (-2 + (-1 + omega3)/(1 - gamma))* (-1 + (-1 + omega3)/(1 - gamma))* (-1 + omega3)* ...
         omega3* theta3* ((1 - gamma)* w(wnStart:Nn)).^(-3 + (-1 + omega3)/(1 - gamma));
     
     
     
    DD(wnStart:Nn)=mu1*yy(wnStart:Nn).^beta1+mu2*yy(wnStart:Nn).^beta2;
    
    dDDdw(wnStart:Nn)=beta1* mu1* ((1 - gamma)* w(wnStart:Nn)).^(-1 + beta1/(1 - gamma)) + ... 
        beta2* mu2* ((1 - gamma)* w(wnStart:Nn)).^(-1 + beta2/(1 - gamma));
    d2DDdw2(wnStart:Nn)=beta1* (-1 + beta1/(1 - gamma))* (1 - gamma)* mu1* ((1 - gamma)* w(wnStart:Nn)).^(-2 + beta1/(1 - gamma)) + ...
        beta2* (-1 + beta2/(1 - gamma))* (1 - gamma)* mu2* ((1 - gamma)* w(wnStart:Nn)).^(-2 + beta2/(1 - gamma));
    
    dGG2(wnStart:Nn)=theta1*omega1*(omega1-1).*yy(wnStart:Nn).^(omega1-2)+ ...
        theta2*omega2*(omega2-1).*yy(wnStart:Nn).^(omega2-2)+ ...
       theta3*omega3*(omega3-1).*yy(wnStart:Nn).^(omega3-2);
    
    ddGG2dw(wnStart:Nn)=(-2+omega1)* (-1+omega1)* omega1* theta1 *((1-gamma)* w(wnStart:Nn)).^(-1+(-2+omega1)/(1-gamma))+ ...
        (-2+omega2)* (-1+omega2)* omega2* theta2* ((1-gamma)* w(wnStart:Nn)).^(-1+(-2+omega2)/(1-gamma))+ ...
        (-2+omega3)* (-1+omega3)* omega3* theta3* ((1-gamma)* w(wnStart:Nn)).^(-1+(-2+omega3)/(1-gamma));
    
    d2dGG2dw2(wnStart:Nn)=(1 - gamma)* (-1 + (-2 + omega1)/(1 - gamma))* (-2 + omega1)* ...
        (-1 + omega1)* omega1* theta1* ((1 - gamma)* w(wnStart:Nn)).^(-2 + (-2 + omega1)/(1 - gamma)) + ...
        (1 - gamma)* (-1 + (-2 + omega2)/(1 - gamma))* (-2 + omega2)* (-1 + omega2)* omega2 * ...
        theta2* ((1 - gamma)* w(wnStart:Nn)).^(-2 + (-2 + omega2)/(1 - gamma)) + ...
        (1 - gamma)*(-1 + (-2 + omega3)/(1 - gamma))* (-2 + omega3)* (-1 + omega3)* omega3 * ...
        theta3* ((1 - gamma)* w(wnStart:Nn)).^(-2 + (-2 + omega3)/(1 - gamma));
    
    QQ(wnStart:Nn)=sigma0^2*yy(wnStart:Nn).^(2*gamma);
    
    dQQdw(wnStart:Nn)=2* gamma* sigma0^2* ((1 - gamma)* w(wnStart:Nn)).^(-1 + (2 *gamma)/(1 - gamma));
    
    d2QQdw2(wnStart:Nn)=2* (1 - gamma)* gamma* (-1 + (2* gamma)/(1 - gamma)) * ...
        sigma0^2* ((1 - gamma)* w(wnStart:Nn)).^(-2 + (2* gamma)/(1 - gamma));
    
    ddGGDD(wnStart:Nn)=mu1*omega1*theta1*(omega1+beta1-1).*yy(wnStart:Nn).^(omega1+beta1-2)+ ...
        mu1*omega2*theta2*(omega2+beta1-1).*yy(wnStart:Nn).^(omega2+beta1-2)+ ...
        mu1*omega3*theta3*(omega3+beta1-1).*yy(wnStart:Nn).^(omega3+beta1-2)+ ...
        mu2*omega1*theta1*(omega1+beta2-1).*yy(wnStart:Nn).^(omega1+beta2-2)+ ...
        mu2*omega2*theta2*(omega2+beta2-1).*yy(wnStart:Nn).^(omega2+beta2-2)+ ...
        mu2*omega3*theta3*(omega3+beta2-1).*yy(wnStart:Nn).^(omega3+beta2-2);
    
    d2dGGDD(wnStart:Nn)=mu1*omega1*theta1*(omega1+beta1-1).*(omega1+beta1-2).*yy(wnStart:Nn).^(omega1+beta1-3)+ ...
        mu1*omega2*theta2*(omega2+beta1-1).*(omega2+beta1-2).*yy(wnStart:Nn).^(omega2+beta1-3)+ ...
        mu1*omega3*theta3*(omega3+beta1-1).*(omega3+beta1-2).*yy(wnStart:Nn).^(omega3+beta1-3)+ ...
        mu2*omega1*theta1*(omega1+beta2-1).*(omega1+beta2-2).*yy(wnStart:Nn).^(omega1+beta2-3)+ ...
        mu2*omega2*theta2*(omega2+beta2-1).*(omega2+beta2-2).*yy(wnStart:Nn).^(omega2+beta2-3)+ ...
        mu2*omega3*theta3*(omega3+beta2-1).*(omega3+beta2-2).*yy(wnStart:Nn).^(omega3+beta2-3);

    ddGG2QQ(wnStart:Nn)=theta1*sigma0^2.*omega1*(omega1-1).*(omega1+2*gamma-2).*yy(wnStart:Nn).^(omega1+2*gamma-3)+ ...
        sigma0^2*theta2*omega2*(omega2-1).*(omega2+2*gamma-2).*yy(wnStart:Nn).^(omega2+2*gamma-3)+ ...
        sigma0^2*theta3*omega3*(omega3-1).*(omega3+2*gamma-2).*yy(wnStart:Nn).^(omega3+2*gamma-3);
        
    d2dGG2QQ(wnStart:Nn)=theta1*sigma0^2.*omega1*(omega1-1).*(omega1+2*gamma-2).*(omega1+2*gamma-3).*yy(wnStart:Nn).^(omega1+2*gamma-4)+ ...
        sigma0^2*theta2*omega2*(omega2-1).*(omega2+2*gamma-2).*(omega2+2*gamma-3).*yy(wnStart:Nn).^(omega2+2*gamma-4)+ ...
        sigma0^2*theta3*omega3*(omega3-1).*(omega3+2*gamma-2).*(omega3+2*gamma-3).*yy(wnStart:Nn).^(omega3+2*gamma-4);
    
    VV(wnStart:Nn)=sigma0.*yy(wnStart:Nn).^gamma;
    d_dGG_VV_dyy(wnStart:Nn)= ...
        sigma0.*omega1.*theta1.*(omega1+gamma-1).*yy(wnStart:Nn).^(omega1+gamma-2) + ...
        sigma0.*omega2.*theta2.*(omega2+gamma-1).*yy(wnStart:Nn).^(omega2+gamma-2) + ...
        sigma0.*omega3.*theta3.*(omega3+gamma-1).*yy(wnStart:Nn).^(omega3+gamma-2);    

    wMu0dt(wnStart:Nn)=GG(wnStart:Nn)*dt + dGG(wnStart:Nn).*DD(wnStart:Nn).*dt2 + ...
        .5*dGG2(wnStart:Nn).*QQ(wnStart:Nn).*dt2 + ...
        ddGGDD(wnStart:Nn).*DD(wnStart:Nn).*dt3 + ...
        .5*ddGG2QQ(wnStart:Nn).*DD(wnStart:Nn).*dt3 + ...
        .5*d2dGGDD(wnStart:Nn).*QQ(wnStart:Nn).*dt3 + ...
        .25*d2dGG2QQ(wnStart:Nn).*QQ(wnStart:Nn).*dt3;
    dwMu0dtdw(wnStart:Nn)=dGGdw(wnStart:Nn)*dt + ddGGdw(wnStart:Nn).*DD(wnStart:Nn).*dt2 + ...
        dGG(wnStart:Nn).*dDDdw(wnStart:Nn).*dt2+ ...
        .5*ddGG2dw(wnStart:Nn).*QQ(wnStart:Nn).*dt2 + ...
        .5*dGG2(wnStart:Nn).*dQQdw(wnStart:Nn).*dt2;
    d2wMu0dtdw2(wnStart:Nn)=d2GGdw2(wnStart:Nn)*dt + d2dGGdw2(wnStart:Nn).*DD(wnStart:Nn).*dt2 + ...
        2*ddGGdw(wnStart:Nn).*dDDdw(wnStart:Nn).*dt2+ ...
        dGG(wnStart:Nn).*d2DDdw2(wnStart:Nn).*dt2+ ...
        .5*d2dGG2dw2(wnStart:Nn).*QQ(wnStart:Nn).*dt2 + ...
        .5*2*ddGG2dw(wnStart:Nn).*dQQdw(wnStart:Nn).*dt2 + ...
        .5*dGG2(wnStart:Nn).*d2QQdw2(wnStart:Nn).*dt2;
    
    d3wMu0dtdw3(wnStart:Nn)=d3GGdw3(wnStart:Nn)*dt;
    
       
    %    c0(wnStart:Nn)=-1*d_dGG_VV_dyy(wnStart:Nn).*VV(wnStart:Nn).*(1/sqrt(2.0).*(1-sqrt(2.0)/2.0)).*dt.^2.0;
    c1(wnStart:Nn)=sigma0.*dtsqrt+dGG(wnStart:Nn).*VV(wnStart:Nn).*(1-1/sqrt(3.0)).*dtonehalf;
    c2(wnStart:Nn)=d_dGG_VV_dyy(wnStart:Nn).*VV(wnStart:Nn).*(1/sqrt(2.0).*(1-sqrt(2.0)/2.0)).*dt.^2.0;
%    dw(wnStart:Nn)=c1(wnStart:Nn).*Z(wnStart:Nn);% + ...
%    dw2(wnStart:Nn)=c1(wnStart:Nn).^2.0.*Z(wnStart:Nn).^2;
    

end

Here is the 2nd function required.

function [C0,C1,C2,w0,C20,wMean,ZMean,nnMean,Cm1,Cm2] = CalculateHermitesOrderTwo(w,Z,Prob,wnStart,Nn,dNn,NnMidh,NnMidl)

%-------------------------------------------------------------------------
%Below hermite coefficients calculation starts. InterpolateOrderN is a
%program that interpolates the values of w0 at Z=0 since in our set up grid
%points are located at Z(NnMidl)=-.1 and Z(NnMidh)=.1 and there are no grid
%points at Z=0.

Z0=0;
w0= InterpolateOrderN(4,Z0,Z(NnMidh),Z(NnMidh+1),Z(NnMidl),Z(NnMidl-1),0,w(NnMidh),w(NnMidh+1),w(NnMidl),w(NnMidl-1),0);
dwdZ0=(w(NnMidh)-w(NnMidl))./dNn;
d2wdZ20 = (-1*w(NnMidl-1)+81*w(NnMidl)-160*w0+81*w(NnMidh)-1*w(NnMidh+1))/(72*(.5*dNn)^2);

%Calculate first hermite polynomial coefficients C1 below.
dwdZ(wnStart)=(-3*w(wnStart)+4*w(wnStart+1)-1*w(wnStart+2))/(2*dNn);
dwdZ(wnStart+1:Nn-1)=(-1*w(wnStart:Nn-2)+1*w(wnStart+2:Nn))/(2*dNn);
dwdZ(Nn)=(3*w(Nn)-4*w(Nn-1)+1*w(Nn-2))/(2*dNn);

C1(wnStart:Nn)=0;
C1(NnMidh)=.5*dwdZ0+.5*dwdZ(NnMidh);
for nn=NnMidh+1:Nn
        C1(nn)=(C1(nn-1).*Z(nn-1)+(.5*dwdZ(nn-1)+.5*dwdZ(nn))*dNn)./Z(nn);
end
C1(NnMidl)=.5*dwdZ0+.5*dwdZ(NnMidl);
for nn=NnMidl-1:-1:wnStart
        C1(nn)=(C1(nn+1).*Z(nn+1)-(.5*dwdZ(nn+1)+.5*dwdZ(nn))*dNn)./Z(nn);
end

%Calculate second hermite coefficients below. Here C20 are coefficients of
%second hermite polynomial at Z=0 
C20=.5*d2wdZ20;
C0=w0+C20;

for nn=NnMidh:Nn
    C2(nn)=((w(nn)-w0)-C1(nn).*Z(nn)-C20)./(Z(nn).^2-1);
end
for nn=NnMidl:-1:wnStart
    C2(nn)=((w(nn)-w0)-C1(nn).*Z(nn)-C20)./(Z(nn).^2-1);
end

w2(wnStart:Nn)=C0+C1(wnStart:Nn).*Z(wnStart:Nn)+C2(wnStart:Nn).*(Z(wnStart:Nn).^2-1);



%Mean is calculated and returned but it has nothing to do with calculation
%of hermite coefficients. It is for future use.
%--------------------------------------------------------------------
wMean=sum(w(wnStart:Nn).*Prob(wnStart:Nn));  %Calculate the mean. This is in general 

%nnMean=NnMidl;
%for nn=wnStart:Nn-1
%    if( (w(nn+1)>wMean) && (w(nn)<=wMean))
%        %ZMean=Z(nn)+(wMean-w(nn))/(w(nn+1)-w(nn))*(Z(nn+1)-Z(nn));
%        nnMean=nn;
%    end
%end

MeanFoundFlag=0;
nnMean=NnMidl;
for nn=NnMidl:Nn-1
    if( (w(nn+1)>wMean) && (w(nn)<=wMean)&& (MeanFoundFlag==0))
        %ZMean=Z(nn)+(wMean-w(nn))/(w(nn+1)-w(nn))*(Z(nn+1)-Z(nn));
        nnMean=nn;
        MeanFoundFlag=1;
    end
end

for nn=NnMidl-1:-1:wnStart
    if( (w(nn+1)>wMean) && (w(nn)<=wMean)&& (MeanFoundFlag==0))
        %ZMean=Z(nn)+(wMean-w(nn))/(w(nn+1)-w(nn))*(Z(nn+1)-Z(nn));
        nnMean=nn;
        MeanFoundFlag=1;
    end
end



if(abs(wMean-w(nnMean))<abs(wMean-w(nnMean+1)))

ZMean= InterpolateOrderN(5,wMean,w(nnMean-2),w(nnMean-1),w(nnMean),w(nnMean+1),w(nnMean+2),Z(nnMean-2),Z(nnMean-1),Z(nnMean),Z(nnMean+1),Z(nnMean+2));

end

if(abs(wMean-w(nnMean))>=abs(wMean-w(nnMean+1)))

ZMean= InterpolateOrderN(5,wMean,w(nnMean-1),w(nnMean),w(nnMean+1),w(nnMean+2),w(nnMean+3),Z(nnMean-1),Z(nnMean),Z(nnMean+1),Z(nnMean+2),Z(nnMean+3));
end


if ((ZMean>=0)&&(ZMean<dNn/4))
    Cm1=dwdZ0;
end
if ((ZMean>dNn/4)&&(ZMean<dNn/2))
    Cm1=(dwdZ0*dNn/4+(ZMean-dNn/4)*dwdZ(NnMidh))/ZMean;
end

if(ZMean>Z(NnMidh))
    if(ZMean<Z(nnMean)+dNn/2)
        Cm1=(C1(nnMean).*Z(nnMean)+dwdZ(nnMean).*(ZMean-Z(nnMean)))./ZMean;
    end
    if(ZMean>Z(nnMean)+dNn/2)
        Cm1=(C1(nnMean).*Z(nnMean)+dwdZ(nnMean).*dNn/2+dwdZ(nnMean+1).*(ZMean-Z(nnMean)-dNn/2))./ZMean;
    end
end

if ((ZMean<0)&&(ZMean>-dNn/4))
    Cm1=dwdZ0;
end
if ((ZMean<-dNn/4)&&(ZMean>-dNn/2))
    Cm1=(dwdZ0*-dNn/4+(ZMean+dNn/4)*dwdZ(NnMidl))/ZMean;
end

if(ZMean<Z(NnMidl))
    if(ZMean>Z(nnMean+1)-dNn/2)
        Cm1=(C1(nnMean+1).*Z(nnMean+1)+dwdZ(nnMean).*(ZMean-Z(nnMean+1)))./ZMean;
    end
    if(ZMean<Z(nnMean)-dNn/2)
        Cm1=(C1(nnMean+1).*Z(nnMean+1)+dwdZ(nnMean+1).*-dNn/2+dwdZ(nnMean).*(ZMean-Z(nnMean+1)+dNn/2))./ZMean;
    end
end

    
Cm2=((wMean-w0)-Cm1.*ZMean-C20)./(ZMean.^2-1);






%Below we plot a graphical comparison of the original SDE variable w and
%the variable w1 reproduced from the coefficients of the hermite
%polynomials
% plot(Z(wnStart:Nn),w(wnStart:Nn),'g',Z(wnStart:Nn),w2(wnStart:Nn),'r')
%Sigma00

C1
C2
C0
% str=input('Look at the graphical comparison of w and w1 as a function of standard gaussian Z');
end

Here is the third required function.

function [C0,C1,C2,w0,C20] = CalculateHermitesOrderTwoWithoutMean(w,Z,Prob,wnStart,Nn,dNn,NnMidh,NnMidl)

%-------------------------------------------------------------------------
%Below hermite coefficients calculation starts. InterpolateOrderN is a
%program that interpolates the values of w0 at Z=0 since in our set up grid
%points are located at Z(NnMidl)=-.1 and Z(NnMidh)=.1 and there are no grid
%points at Z=0.

Z0=0;
w0= InterpolateOrderN(4,Z0,Z(NnMidh),Z(NnMidh+1),Z(NnMidl),Z(NnMidl-1),0,w(NnMidh),w(NnMidh+1),w(NnMidl),w(NnMidl-1),0);
dwdZ0=(w(NnMidh)-w(NnMidl))./dNn;
d2wdZ20 = (-1*w(NnMidl-1)+81*w(NnMidl)-160*w0+81*w(NnMidh)-1*w(NnMidh+1))/(72*(.5*dNn)^2);

%Calculate first hermite polynomial coefficients C1 below.
dwdZ(wnStart)=(-3*w(wnStart)+4*w(wnStart+1)-1*w(wnStart+2))/(2*dNn);
dwdZ(wnStart+1:Nn-1)=(-1*w(wnStart:Nn-2)+1*w(wnStart+2:Nn))/(2*dNn);
dwdZ(Nn)=(3*w(Nn)-4*w(Nn-1)+1*w(Nn-2))/(2*dNn);

C1(wnStart:Nn)=0;
C1(NnMidh)=.5*dwdZ0+.5*dwdZ(NnMidh);
for nn=NnMidh+1:Nn
        C1(nn)=(C1(nn-1).*Z(nn-1)+(.5*dwdZ(nn-1)+.5*dwdZ(nn))*dNn)./Z(nn);
end
C1(NnMidl)=.5*dwdZ0+.5*dwdZ(NnMidl);
for nn=NnMidl-1:-1:wnStart
        C1(nn)=(C1(nn+1).*Z(nn+1)-(.5*dwdZ(nn+1)+.5*dwdZ(nn))*dNn)./Z(nn);
end

%Calculate second hermite coefficients below. Here C20 are coefficients of
%second hermite polynomial at Z=0 
C20=.5*d2wdZ20;
C0=w0+C20;

for nn=NnMidh:Nn
    C2(nn)=((w(nn)-w0)-C1(nn).*Z(nn)-C20)./(Z(nn).^2-1);
end
for nn=NnMidl:-1:wnStart
    C2(nn)=((w(nn)-w0)-C1(nn).*Z(nn)-C20)./(Z(nn).^2-1);
end

w2(wnStart:Nn)=C0+C1(wnStart:Nn).*Z(wnStart:Nn)+C2(wnStart:Nn).*(Z(wnStart:Nn).^2-1);



%Mean is calculated and returned but it has nothing to do with calculation
%of hermite coefficients. It is for future use.
%--------------------------------------------------------------------
%wMean=sum(w(wnStart:Nn).*Prob(wnStart:Nn));  %Calculate the mean. This is in general 

%nnMean=NnMidl;
%for nn=wnStart:Nn-1
%    if( (w(nn+1)>wMean) && (w(nn)<=wMean))
%        %ZMean=Z(nn)+(wMean-w(nn))/(w(nn+1)-w(nn))*(Z(nn+1)-Z(nn));
%        nnMean=nn;
%    end
%end




%Below we plot a graphical comparison of the original SDE variable w and
%the variable w1 reproduced from the coefficients of the hermite
%polynomials
% plot(Z(wnStart:Nn),w(wnStart:Nn),'g',Z(wnStart:Nn),w2(wnStart:Nn),'r')
%Sigma00

C1
C2
C0
% str=input('Look at the graphical comparison of w and w1 as a function of standard gaussian Z');
end

Here is the 4th required function.

function [wMean,ZMean,Sigma00,Sigma11,Sigma22,Sigma33,Sigma1,Sigma2,Sigma3,Sigma0,w1,w2,nnMean] = CalculateHermiteRepresentationOfDensityOrderThree(w,Z,Prob,wnStart,Nn,dNn,NnMidh,NnMidl)

%This program find hermite representation of the SDE variable w to 
%tjird Order as a function of three hermite polynomials.
%This program is not perfectly precise and has very small errors so you
%will have to be careful using it everywhere.
%Here wnStart is the starting grid point of the SDE vector
%mostly wnStart=1.
%Please note we are representing SDE Variable w as a function of hermite 
%polynomials. It is not the density which can however be found by change of variable 
%derivative of w with respect to Z using standard gaussian variable.
%Nn is the index of last density grid point.
%So w(wnStart:Nn) represents the SDE density variable grid.
%  Z(wnStart:Nn) represents the associated value of standard Gaussian on
%  the grid. w and Z can be associated by matching appropriate CDF points
%  on the grid.
%I suppose the grid is uniform in Z and dNn is the spacing between and 
% Z(n) and Z(n+1) two points on the grid.
%Prob is the probability mass associated with each point on the grid.

wMean=sum(w(wnStart:Nn).*Prob(wnStart:Nn));  %Calculate the mean. This is in general 
%not on the discrete grid of w. 

%Below calculate the associated value of ZMean that maps wMean to ZMean.
%Also the grid point index nnMean which is associated with the largest
% w(nn) such that w(nn)<=wMean.
ZMean=(w(NnMidh)+w(NnMidl))/2.0;
nnMean=NnMidl;
for nn=wnStart:Nn-1
    if( (w(nn+1)>wMean) && (w(nn)<=wMean))
        ZMean=Z(nn)+(wMean-w(nn))/(w(nn+1)-w(nn))*(Z(nn+1)-Z(nn));
        nnMean=nn;
    end
end

%Below Calculate the vol associated with each point on the grid
for nn=wnStart:Nn
    Sigma0(nn)=(w(nn)-wMean)./(Z(nn)-ZMean);  
 end

%Below calculate the associated Sigma00 which is a single volatility associated with
%the first orthogonal hermite polynomial. We chose this volatility equal to
%volatility at mean since we are expanding around the mean.
%If mean has zero volatility, we will have to find volatility at mean by
%interpolating the volatility at one higher grid point and one lower grid
%point. Volatility at mean must not be zero (unless we have a delta density).
Sigma00=Sigma0(nnMean)*(Z(nnMean+1)-ZMean)./(Z(nnMean+1)-Z(nnMean))+Sigma0(nnMean+1)*(ZMean-Z(nnMean))./(Z(nnMean+1)-Z(nnMean));

%Below calculate the associated first derivative of Sigma0 with respect to
%Z on the grid.
dSigma0dZ(wnStart)=(-3*Sigma0(wnStart)+4*Sigma0(wnStart+1)-1*Sigma0(wnStart+2))/(2*dNn);
dSigma0dZ(wnStart+1:Nn-1)=(-1*Sigma0(wnStart:Nn-2)+1*Sigma0(wnStart+2:Nn))/(2*dNn);
dSigma0dZ(Nn)=(3*Sigma0(Nn)-4*Sigma0(Nn-1)+1*Sigma0(Nn-2))/(2*dNn);

%Following is the operation related to equation(4) in post 860.     
SSigma1(nnMean)=0;
for nn=nnMean+1:Nn
        SSigma1(nn)=SSigma1(nn-1)+(.5*dSigma0dZ(nn-1)+.5*dSigma0dZ(nn))*dNn;
end
for nn=nnMean-1:-1:wnStart
        SSigma1(nn)=SSigma1(nn+1)-(.5*dSigma0dZ(nn+1)+.5*dSigma0dZ(nn))*dNn;
end

%Below is the averaging
Sigma1(wnStart:nnMean-1)=SSigma1(wnStart:nnMean-1)./(Z(wnStart:nnMean-1)-Z(nnMean));
Sigma1(nnMean+1:Nn)=SSigma1(nnMean+1:Nn)./(Z(nnMean+1:Nn)-Z(nnMean));
Sigma1(nnMean)=(Sigma1(nnMean-1)+Sigma1(nnMean+1))/2.0;


Sigma11=Sigma1(nnMean)*(Z(nnMean+1)-ZMean)./(Z(nnMean+1)-Z(nnMean))+Sigma1(nnMean+1)*(ZMean-Z(nnMean))./(Z(nnMean+1)-Z(nnMean));



%Now below we represent the SDE variable w in terms of first two hermite
%polynomials and this associated value of w produced by hermite polynomials
%coefficients is called w1.
   
% w1(wnStart:Nn)=wMean-Sigma00*ZMean-Sigma1(wnStart:Nn).*(ZMean.^2-1)+ ...
% Sigma00*Z(wnStart:Nn)+ ...
% Sigma1(wnStart:Nn).*(Z(wnStart:Nn).^2-1);

w0(wnStart:Nn)=wMean-Sigma0(wnStart:Nn)*ZMean+Sigma0(wnStart:Nn).*Z(wnStart:Nn);

w1(wnStart:Nn)=wMean-Sigma00*ZMean-Sigma1(wnStart:Nn).*(ZMean.^2-1)+ ...
 -Sigma1(wnStart:Nn).*Z(wnStart:Nn).*ZMean+ ...   
 Sigma00*Z(wnStart:Nn)+ ...
 Sigma1(wnStart:Nn).*(Z(wnStart:Nn).^2-1);

w2(wnStart:Nn)=wMean-Sigma00*ZMean-Sigma1(wnStart:Nn).*(ZMean.^2-1)+ ...
 -2*Sigma1(wnStart:Nn).*(Z(wnStart:Nn)-ZMean).*ZMean+ ...   
 Sigma00*Z(wnStart:Nn)+ ...
 Sigma1(wnStart:Nn).*(Z(wnStart:Nn).^2-1);
 



dSigma1dZ(wnStart)=(-3*Sigma1(wnStart)+4*Sigma1(wnStart+1)-1*Sigma1(wnStart+2))/(2*dNn);
dSigma1dZ(wnStart+1:Nn-1)=(-1*Sigma1(wnStart:Nn-2)+1*Sigma1(wnStart+2:Nn))/(2*dNn);
dSigma1dZ(Nn)=(3*Sigma1(Nn)-4*Sigma1(Nn-1)+1*Sigma1(Nn-2))/(2*dNn);

% Sigma2(NnMidh)=dSigma1dZ(NnMidh);
% for nn=NnMidh+1:Nn
%     Sigma2(nn)=(Sigma2(nn-1)*Z(nn-1)+(.5*dSigma1dZ(nn-1)+.5*dSigma1dZ(nn))*dNn)./Z(nn);
% end
% Sigma2(NnMidl)=dSigma1dZ(NnMidl);
% for nn=NnMidl-1:-1:wnStart
%     Sigma2(nn)=(Sigma2(nn+1)*Z(nn+1)-(.5*dSigma1dZ(nn+1)+.5*dSigma1dZ(nn))*dNn)./Z(nn);
% end

SSigma2(nnMean)=0;
for nn=nnMean+1:Nn
        SSigma2(nn)=SSigma2(nn-1)+(.5*dSigma1dZ(nn-1)+.5*dSigma1dZ(nn))*dNn;
end
for nn=nnMean-1:-1:wnStart
        SSigma2(nn)=SSigma2(nn+1)-(.5*dSigma1dZ(nn+1)+.5*dSigma1dZ(nn))*dNn;
end

Sigma2(wnStart:nnMean-1)=SSigma2(wnStart:nnMean-1)./(Z(wnStart:nnMean-1)-Z(nnMean));
Sigma2(nnMean+1:Nn)=SSigma2(nnMean+1:Nn)./(Z(nnMean+1:Nn)-Z(nnMean));
Sigma2(nnMean)=(Sigma2(nnMean-1)+Sigma2(nnMean+1))/2.0;

%Sigma22=Sigma2(nnMean);
Sigma22=Sigma2(nnMean)*(Z(nnMean+1)-ZMean)./(Z(nnMean+1)-Z(nnMean))+Sigma2(nnMean+1)*(ZMean-Z(nnMean))./(Z(nnMean+1)-Z(nnMean));

dSigma2dZ(wnStart)=(-3*Sigma2(wnStart)+4*Sigma2(wnStart+1)-1*Sigma2(wnStart+2))/(2*dNn);
dSigma2dZ(wnStart+1:Nn-1)=(-1*Sigma2(wnStart:Nn-2)+1*Sigma2(wnStart+2:Nn))/(2*dNn);
dSigma2dZ(Nn)=(3*Sigma2(Nn)-4*Sigma2(Nn-1)+1*Sigma2(Nn-2))/(2*dNn);


SSigma3(nnMean)=0;
for nn=nnMean+1:Nn
        SSigma3(nn)=SSigma3(nn-1)+(.5*dSigma2dZ(nn-1)+.5*dSigma2dZ(nn))*dNn;
end
for nn=nnMean-1:-1:wnStart
        SSigma3(nn)=SSigma3(nn+1)-(.5*dSigma2dZ(nn+1)+.5*dSigma2dZ(nn))*dNn;
end

Sigma3(wnStart:nnMean-1)=SSigma3(wnStart:nnMean-1)./(Z(wnStart:nnMean-1)-Z(nnMean));
Sigma3(nnMean+1:Nn)=SSigma3(nnMean+1:Nn)./(Z(nnMean+1:Nn)-Z(nnMean));
Sigma3(nnMean)=(Sigma3(nnMean-1)+Sigma3(nnMean+1))/2.0;

Sigma33=Sigma3(nnMean)*(Z(nnMean+1)-ZMean)./(Z(nnMean+1)-Z(nnMean))+Sigma3(nnMean+1)*(ZMean-Z(nnMean))./(Z(nnMean+1)-Z(nnMean));




% w2(wnStart:Nn)=wMean-Sigma00*ZMean-Sigma11.*(ZMean.^2-1)- ...
%  Sigma2(wnStart:Nn).*(ZMean.^3-3*ZMean) - ...
%  0*3*Sigma2(wnStart:Nn).*Z(wnStart:Nn).*(ZMean.^2-1) + ... 
%  Sigma00*Z(wnStart:Nn)+ ...
%  Sigma11.*(Z(wnStart:Nn).^2-1)+ ...   
%  Sigma2(wnStart:Nn).*(Z(wnStart:Nn).^3-3*Z(wnStart:Nn));
 



%Below we plot a graphical comparison of the original SDE variable w and
%the variable w1 reproduced from the coefficients of the hermite
%polynomials
% plot(Z(wnStart:Nn),w(wnStart:Nn),'g',Z(wnStart:Nn),w1(wnStart:Nn),'r',Z(wnStart:Nn),w2(wnStart:Nn),'b',Z(wnStart:Nn),w0(wnStart:Nn),'k')
%Sigma00
% str=input('Look at the graphical comparison of w and w1 as a function of standard gaussian Z');
end

Here is another small function required

function [y0] = InterpolateOrderN(N,x0,x1,x2,x3,x4,x5,y1,y2,y3,y4,y5)


if(N==5)
y0=(x0-x2)*(x0-x3)*(x0-x4)*(x0-x5)/((x1-x2).*(x1-x3).*(x1-x4).*(x1-x5))*y1+ ...
   (x0-x1)*(x0-x3)*(x0-x4)*(x0-x5)/((x2-x1).*(x2-x3).*(x2-x4).*(x2-x5))*y2+ ...
   (x0-x1)*(x0-x2)*(x0-x4)*(x0-x5)/((x3-x1).*(x3-x2).*(x3-x4).*(x3-x5))*y3+ ...
   (x0-x1)*(x0-x2)*(x0-x3)*(x0-x5)/((x4-x1).*(x4-x2).*(x4-x3).*(x4-x5))*y4+ ...
   (x0-x1)*(x0-x2)*(x0-x3)*(x0-x4)/((x5-x1).*(x5-x2).*(x5-x3).*(x5-x4))*y5;
end
if(N==4)
y0=(x0-x2)*(x0-x3)*(x0-x4)/((x1-x2).*(x1-x3).*(x1-x4))*y1+ ...
   (x0-x1)*(x0-x3)*(x0-x4)/((x2-x1).*(x2-x3).*(x2-x4))*y2+ ...
   (x0-x1)*(x0-x2)*(x0-x4)/((x3-x1).*(x3-x2).*(x3-x4))*y3+ ...
   (x0-x1)*(x0-x2)*(x0-x3)/((x4-x1).*(x4-x2).*(x4-x3))*y4;
end
if(N==3)
y0=(x0-x2)*(x0-x3)/((x1-x2).*(x1-x3))*y1+ ...
   (x0-x1)*(x0-x3)/((x2-x1).*(x2-x3))*y2+ ...
   (x0-x1)*(x0-x2)/((x3-x1).*(x3-x2))*y3;
end
if(N==2)
y0=(x0-x2)/((x1-x2))*y1+ ...
   (x0-x1)/((x2-x1))*y2;
end

%y0=(x0-x3)/(x2-x3)*y2+(x0-x2)/(x3-x2)*y3;

end

I am not including the dependent functions for Monte carlo as they are the same as I have already previously posted.

Amin · March 13th, 2020, 5:08 pm

Here is a new version of the program that calculates hermite representation. It is still not extended to order three but it has better interpolations and better calculations of derivatives. With this new program that effect of instability in the middle of the density has sharply decreased but it still does remain for higher values of beta after .5 years. At T=.5 years, the effect of instability is extremely small but for higher values of beta, it is visible in the graph and is non-negligible. I will try to post another version with third order hermites probably sometime later tonight. Here is improved version with two hermite polynomials.
In order to see how the instability has decreased, you can run the previous program with this improved function and make fudge constant before the main update equation equal to zero to see the effect of instability after commenting the instability correction part of the program.

function [C0,C1,C2,w0,C20,wMean,ZMean,nnMean,Cm1,Cm2] = CalculateHermitesOrderTwo(w,Z,Prob,wnStart,Nn,dNn,NnMidh,NnMidl)

%-------------------------------------------------------------------------
%Below hermite coefficients calculation starts. InterpolateOrderN is a
%program that interpolates the values of w0 at Z=0 since in our set up grid
%points are located at Z(NnMidl)=-.1 and Z(NnMidh)=.1 and there are no grid
%points at Z=0.

Z0=0;
%w0= InterpolateOrderN(4,Z0,Z(NnMidh),Z(NnMidh+1),Z(NnMidl),Z(NnMidl-1),0,w(NnMidh),w(NnMidh+1),w(NnMidl),w(NnMidl-1),0);
w0= InterpolateOrderN6(6,Z0,Z(NnMidh+2),Z(NnMidh+1),Z(NnMidh),Z(NnMidl),Z(NnMidl-1),Z(NnMidl-2),w(NnMidh+2),w(NnMidh+1),w(NnMidh),w(NnMidl),w(NnMidl-1),w(NnMidl-2));
%dwdZ0=(w(NnMidh)-w(NnMidl))./dNn;
%dwdZ0= (1*w(NnMidl-1)-27*w(NnMidl)+27*w(NnMidh)-1*w(NnMidh+1))/(48*(.5*dNn));
dwdZ0 = (-9*w(NnMidl-2)+125*w(NnMidl-1)-2250*w(NnMidl)+2250*w(NnMidh)-125*w(NnMidh+1)+9*w(NnMidh+2))/(3840*(.5*dNn));
%d2wdZ20 = (-1*w(NnMidl-1)+81*w(NnMidl)-160*w0+81*w(NnMidh)-1*w(NnMidh+1))/(72*(.5*dNn)^2);
%d2wdZ20 = (-1*w(NnMidl-1)+81*w(NnMidl)-160*w0+81*w(NnMidh)-1*w(NnMidh+1))/(72*(.5*dNn)^2)
d2wdZ20 = (-5*w(NnMidl-2)+39*w(NnMidl-1)-34*w(NnMidl)-34*w(NnMidh)+39*w(NnMidh+1)-5*w(NnMidh+2))/(192*(.5*dNn)^2);


%d4wdZ40 = (-3*w(NnMidl-2)+65*w(NnMidl-1)-510*w(NnMidl)+896*w0-510*w(NnMidh)+65*w(NnMidh+1)-3*w(NnMidh+2))/(240*(.5*dNn)^2)

d4wdZ40 = (1*w(NnMidl-2)-3*w(NnMidl-1)+2*w(NnMidl)+2*w(NnMidh)-3*w(NnMidh+1)+1*w(NnMidh+2))/(32*(.5*dNn)^2);

%str=input('Look at derivatives at zero');



%Calculate first hermite polynomial coefficients C1 below.
dwdZ(wnStart)=(-3*w(wnStart)+4*w(wnStart+1)-1*w(wnStart+2))/(2*dNn);
dwdZ(wnStart+1:Nn-1)=(-1*w(wnStart:Nn-2)+1*w(wnStart+2:Nn))/(2*dNn);
dwdZ(Nn)=(3*w(Nn)-4*w(Nn-1)+1*w(Nn-2))/(2*dNn);
dwdZ(wnStart+2:Nn-2)= (1*w(wnStart:Nn-4)-8*w(wnStart+1:Nn-3)+0*w(wnStart+2:Nn-2)+8*w(wnStart+3:Nn-1)-1*w(wnStart+4:Nn))/(12*dNn);

CSum(wnStart:Nn)=0.0;
C1(wnStart:Nn)=0;
C1(NnMidh)=.5*dwdZ0+.5*dwdZ(NnMidh);
CSum(NnMidh)=C1(NnMidh).*Z(NnMidh);
for nn=NnMidh+1:Nn
        CSum(nn)=(C1(nn-1).*Z(nn-1)+(.5*dwdZ(nn-1)+.5*dwdZ(nn))*dNn);
        C1(nn)=CSum(nn)./Z(nn);
end
C1(NnMidl)=.5*dwdZ0+.5*dwdZ(NnMidl);
CSum(NnMidl)=C1(NnMidl).*Z(NnMidl);
for nn=NnMidl-1:-1:wnStart
        CSum(nn)=(C1(nn+1).*Z(nn+1)-(.5*dwdZ(nn+1)+.5*dwdZ(nn))*dNn);
        C1(nn)=CSum(nn)./Z(nn);
end

%Calculate second hermite coefficients below. Here C20 are coefficients of
%second hermite polynomial at Z=0 
C20=.5*d2wdZ20;
C40=1/24*d4wdZ40;
C0=w0+C20-0*3*C40;

for nn=NnMidh:Nn
    C2(nn)=((w(nn)-w0)-C1(nn).*Z(nn)-C20)./(Z(nn).^2-1);
end
for nn=NnMidl:-1:wnStart
    C2(nn)=((w(nn)-w0)-C1(nn).*Z(nn)-C20)./(Z(nn).^2-1);
end

w2(wnStart:Nn)=C0+C1(wnStart:Nn).*Z(wnStart:Nn)+C2(wnStart:Nn).*(Z(wnStart:Nn).^2-1);



%Mean is calculated and returned but it has nothing to do with calculation
%of hermite coefficients. It is for future use.
%--------------------------------------------------------------------
wMean=sum(w(wnStart:Nn).*Prob(wnStart:Nn));  %Calculate the mean. This is in general 

%nnMean=NnMidl;
%for nn=wnStart:Nn-1
%    if( (w(nn+1)>wMean) && (w(nn)<=wMean))
%        %ZMean=Z(nn)+(wMean-w(nn))/(w(nn+1)-w(nn))*(Z(nn+1)-Z(nn));
%        nnMean=nn;
%    end
%end

MeanFoundFlag=0;
nnMean=NnMidl;
for nn=NnMidl:Nn-1
    if( (w(nn+1)>wMean) && (w(nn)<=wMean)&& (MeanFoundFlag==0))
        %ZMean=Z(nn)+(wMean-w(nn))/(w(nn+1)-w(nn))*(Z(nn+1)-Z(nn));
        nnMean=nn;
        MeanFoundFlag=1;
    end
end

for nn=NnMidl-1:-1:wnStart
    if( (w(nn+1)>wMean) && (w(nn)<=wMean)&& (MeanFoundFlag==0))
        %ZMean=Z(nn)+(wMean-w(nn))/(w(nn+1)-w(nn))*(Z(nn+1)-Z(nn));
        nnMean=nn;
        MeanFoundFlag=1;
    end
end



 if(abs(wMean-w(nnMean))<abs(wMean-w(nnMean+1)))
 
 ZMean= InterpolateOrderN(5,wMean,w(nnMean-2),w(nnMean-1),w(nnMean),w(nnMean+1),w(nnMean+2),Z(nnMean-2),Z(nnMean-1),Z(nnMean),Z(nnMean+1),Z(nnMean+2));
 CSum=InterpolateOrderN(5,wMean,w(nnMean-2),w(nnMean-1),w(nnMean),w(nnMean+1),w(nnMean+2),CSum(nnMean-2),CSum(nnMean-1),CSum(nnMean),CSum(nnMean+1),CSum(nnMean+2));
 end
 
 if(abs(wMean-w(nnMean))>=abs(wMean-w(nnMean+1)))
 
 ZMean= InterpolateOrderN(5,wMean,w(nnMean-1),w(nnMean),w(nnMean+1),w(nnMean+2),w(nnMean+3),Z(nnMean-1),Z(nnMean),Z(nnMean+1),Z(nnMean+2),Z(nnMean+3));
 CSum=InterpolateOrderN(5,wMean,w(nnMean-1),w(nnMean),w(nnMean+1),w(nnMean+2),w(nnMean+3),CSum(nnMean-1),CSum(nnMean),CSum(nnMean+1),CSum(nnMean+2),CSum(nnMean+3));
 end
%ZMean=InterpolateOrderN(4,wMean,w(nnMean-1),w(nnMean),w(nnMean+1),w(nnMean+2),0,Z(nnMean-1),Z(nnMean),Z(nnMean+1),Z(nnMean+2),0);
%CSum=InterpolateOrderN(4,wMean,w(nnMean-1),w(nnMean),w(nnMean+1),w(nnMean+2),0,CSum(nnMean-1),CSum(nnMean),CSum(nnMean+1),CSum(nnMean+2),0);


Cm1=CSum./ZMean;


% if ((ZMean>=0)&&(ZMean<dNn/4))
%     Cm1=dwdZ0;
% end
% if ((ZMean>dNn/4)&&(ZMean<dNn/2))
%     Cm1=(dwdZ0*dNn/4+(ZMean-dNn/4)*dwdZ(NnMidh))/ZMean;
% end
% 
% if(ZMean>Z(NnMidh))
%     if(ZMean<Z(nnMean)+dNn/2)
%         Cm1=(C1(nnMean).*Z(nnMean)+dwdZ(nnMean).*(ZMean-Z(nnMean)))./ZMean;
%     end
%     if(ZMean>Z(nnMean)+dNn/2)
%         Cm1=(C1(nnMean).*Z(nnMean)+dwdZ(nnMean).*dNn/2+dwdZ(nnMean+1).*(ZMean-Z(nnMean)-dNn/2))./ZMean;
%     end
% end
% 
% if ((ZMean<0)&&(ZMean>-dNn/4))
%     Cm1=dwdZ0;
% end
% if ((ZMean<-dNn/4)&&(ZMean>-dNn/2))
%     Cm1=(dwdZ0*-dNn/4+(ZMean+dNn/4)*dwdZ(NnMidl))/ZMean;
% end
% 
% if(ZMean<Z(NnMidl))
%     if(ZMean>Z(nnMean+1)-dNn/2)
%         Cm1=(C1(nnMean+1).*Z(nnMean+1)+dwdZ(nnMean).*(ZMean-Z(nnMean+1)))./ZMean;
%     end
%     if(ZMean<Z(nnMean)-dNn/2)
%         Cm1=(C1(nnMean+1).*Z(nnMean+1)+dwdZ(nnMean+1).*-dNn/2+dwdZ(nnMean).*(ZMean-Z(nnMean+1)+dNn/2))./ZMean;
%     end
% end

    
Cm2=((wMean-w0)-Cm1.*ZMean-C20)./(ZMean.^2-1);






%Below we plot a graphical comparison of the original SDE variable w and
%the variable w1 reproduced from the coefficients of the hermite
%polynomials
% plot(Z(wnStart:Nn),w(wnStart:Nn),'g',Z(wnStart:Nn),w2(wnStart:Nn),'r')
%Sigma00

C1
C2
C0
% str=input('Look at the graphical comparison of w and w1 as a function of standard gaussian Z');
end

You will also need the following matlab function.

function [y0] = InterpolateOrderN6(N,x0,x1,x2,x3,x4,x5,x6,y1,y2,y3,y4,y5,y6)

if(N==6)
y0=(x0-x2)*(x0-x3)*(x0-x4)*(x0-x5)*(x0-x6)/((x1-x2).*(x1-x3).*(x1-x4).*(x1-x5).*(x1-x6))*y1+ ...
   (x0-x1)*(x0-x3)*(x0-x4)*(x0-x5)*(x0-x6)/((x2-x1).*(x2-x3).*(x2-x4).*(x2-x5).*(x2-x6))*y2+ ...
   (x0-x1)*(x0-x2)*(x0-x4)*(x0-x5)*(x0-x6)/((x3-x1).*(x3-x2).*(x3-x4).*(x3-x5).*(x3-x6))*y3+ ...
   (x0-x1)*(x0-x2)*(x0-x3)*(x0-x5)*(x0-x6)/((x4-x1).*(x4-x2).*(x4-x3).*(x4-x5).*(x4-x6))*y4+ ...
   (x0-x1)*(x0-x2)*(x0-x3)*(x0-x4)*(x0-x6)/((x5-x1).*(x5-x2).*(x5-x3).*(x5-x4).*(x5-x6))*y5+ ...
   (x0-x1)*(x0-x2)*(x0-x3)*(x0-x4)*(x0-x5)/((x6-x1).*(x6-x2).*(x6-x3).*(x6-x4).*(x6-x5))*y6;
end

if(N==5)
y0=(x0-x2)*(x0-x3)*(x0-x4)*(x0-x5)/((x1-x2).*(x1-x3).*(x1-x4).*(x1-x5))*y1+ ...
   (x0-x1)*(x0-x3)*(x0-x4)*(x0-x5)/((x2-x1).*(x2-x3).*(x2-x4).*(x2-x5))*y2+ ...
   (x0-x1)*(x0-x2)*(x0-x4)*(x0-x5)/((x3-x1).*(x3-x2).*(x3-x4).*(x3-x5))*y3+ ...
   (x0-x1)*(x0-x2)*(x0-x3)*(x0-x5)/((x4-x1).*(x4-x2).*(x4-x3).*(x4-x5))*y4+ ...
   (x0-x1)*(x0-x2)*(x0-x3)*(x0-x4)/((x5-x1).*(x5-x2).*(x5-x3).*(x5-x4))*y5;
end
if(N==4)
y0=(x0-x2)*(x0-x3)*(x0-x4)/((x1-x2).*(x1-x3).*(x1-x4))*y1+ ...
   (x0-x1)*(x0-x3)*(x0-x4)/((x2-x1).*(x2-x3).*(x2-x4))*y2+ ...
   (x0-x1)*(x0-x2)*(x0-x4)/((x3-x1).*(x3-x2).*(x3-x4))*y3+ ...
   (x0-x1)*(x0-x2)*(x0-x3)/((x4-x1).*(x4-x2).*(x4-x3))*y4;
end
if(N==3)
y0=(x0-x2)*(x0-x3)/((x1-x2).*(x1-x3))*y1+ ...
   (x0-x1)*(x0-x3)/((x2-x1).*(x2-x3))*y2+ ...
   (x0-x1)*(x0-x2)/((x3-x1).*(x3-x2))*y3;
end
if(N==2)
y0=(x0-x2)/((x1-x2))*y1+ ...
   (x0-x1)/((x2-x1))*y2;
end

%y0=(x0-x3)/(x2-x3)*y2+(x0-x2)/(x3-x2)*y3;

end

Amin · March 13th, 2020, 10:37 pm

Here is a new version of the program that calculates hermite representation to third order. The first two polynomials should be good as they are based on derivatives and the residual effect is lumped in the third hermite polynomial so it is not guaranteed to be precise but is meant to recover the value of w on all grid points. It should be straightforward to extend the procedure to third and higher polynomials if the numerical derivatives remain stable.
Here is the code.

function [C0,C1,C2,C3,w0,C20,wMean,ZMean,nnMean,Cm1,Cm2,Cm3] = CalculateHermitesOrderThree(w,Z,Prob,wnStart,Nn,dNn,NnMidh,NnMidl)

%-------------------------------------------------------------------------
%Below hermite coefficients calculation starts. InterpolateOrderN is a
%program that interpolates the values of w0 at Z=0 since in our set up grid
%points are located at Z(NnMidl)=-.1 and Z(NnMidh)=.1 and there are no grid
%points at Z=0.

Z0=0;
%w0= InterpolateOrderN(4,Z0,Z(NnMidh),Z(NnMidh+1),Z(NnMidl),Z(NnMidl-1),0,w(NnMidh),w(NnMidh+1),w(NnMidl),w(NnMidl-1),0);
w0= InterpolateOrderN6(6,Z0,Z(NnMidh+2),Z(NnMidh+1),Z(NnMidh),Z(NnMidl),Z(NnMidl-1),Z(NnMidl-2),w(NnMidh+2),w(NnMidh+1),w(NnMidh),w(NnMidl),w(NnMidl-1),w(NnMidl-2));
%dwdZ0=(w(NnMidh)-w(NnMidl))./dNn;
%dwdZ0= (1*w(NnMidl-1)-27*w(NnMidl)+27*w(NnMidh)-1*w(NnMidh+1))/(48*(.5*dNn));
dwdZ0 = (-9*w(NnMidl-2)+125*w(NnMidl-1)-2250*w(NnMidl)+2250*w(NnMidh)-125*w(NnMidh+1)+9*w(NnMidh+2))/(3840*(.5*dNn));
%d2wdZ20 = (-1*w(NnMidl-1)+81*w(NnMidl)-160*w0+81*w(NnMidh)-1*w(NnMidh+1))/(72*(.5*dNn)^2);
%d2wdZ20 = (-1*w(NnMidl-1)+81*w(NnMidl)-160*w0+81*w(NnMidh)-1*w(NnMidh+1))/(72*(.5*dNn)^2)
d2wdZ20 = (-5*w(NnMidl-2)+39*w(NnMidl-1)-34*w(NnMidl)-34*w(NnMidh)+39*w(NnMidh+1)-5*w(NnMidh+2))/(192*(.5*dNn)^2);


%d4wdZ40 = (-3*w(NnMidl-2)+65*w(NnMidl-1)-510*w(NnMidl)+896*w0-510*w(NnMidh)+65*w(NnMidh+1)-3*w(NnMidh+2))/(240*(.5*dNn)^2)

d4wdZ40 = (1*w(NnMidl-2)-3*w(NnMidl-1)+2*w(NnMidl)+2*w(NnMidh)-3*w(NnMidh+1)+1*w(NnMidh+2))/(32*(.5*dNn)^2);

%str=input('Look at derivatives at zero');



%Calculate first hermite polynomial coefficients C1 below.
dwdZ(wnStart)=(-3*w(wnStart)+4*w(wnStart+1)-1*w(wnStart+2))/(2*dNn);
dwdZ(wnStart+1:Nn-1)=(-1*w(wnStart:Nn-2)+1*w(wnStart+2:Nn))/(2*dNn);
dwdZ(Nn)=(3*w(Nn)-4*w(Nn-1)+1*w(Nn-2))/(2*dNn);
dwdZ(wnStart+2:Nn-2)= (1*w(wnStart:Nn-4)-8*w(wnStart+1:Nn-3)+0*w(wnStart+2:Nn-2)+8*w(wnStart+3:Nn-1)-1*w(wnStart+4:Nn))/(12*dNn);





CSum(wnStart:Nn)=0.0;
C1(wnStart:Nn)=0;
C1(NnMidh)=.5*dwdZ0+.5*dwdZ(NnMidh);
CSum(NnMidh)=C1(NnMidh).*Z(NnMidh);
for nn=NnMidh+1:Nn
        CSum(nn)=(C1(nn-1).*Z(nn-1)+(.5*dwdZ(nn-1)+.5*dwdZ(nn))*dNn);
        C1(nn)=CSum(nn)./Z(nn);
end
C1(NnMidl)=.5*dwdZ0+.5*dwdZ(NnMidl);
CSum(NnMidl)=C1(NnMidl).*Z(NnMidl);
for nn=NnMidl-1:-1:wnStart
        CSum(nn)=(C1(nn+1).*Z(nn+1)-(.5*dwdZ(nn+1)+.5*dwdZ(nn))*dNn);
        C1(nn)=CSum(nn)./Z(nn);
end

%Calculate second hermite coefficients below. Here C20 are coefficients of
%second hermite polynomial at Z=0 
C20=.5*d2wdZ20;
C40=1*1/24*d4wdZ40;
C0=w0+C20-3*C40;


[dwdZ,d2wdZ2,d3wdZ3,d4wdZ4] = First4Derivatives2ndOrderEqSpaced(wnStart,Nn,dNn,w,Z);
d2wdZ2(wnStart+2:Nn-2)= (-1*w(wnStart:Nn-4)+16*w(wnStart+1:Nn-3)-30*w(wnStart+2:Nn-2)+16*w(wnStart+3:Nn-1)-1*w(wnStart+4:Nn))/(12*dNn^2);


C2Sum1(wnStart:Nn)=0.0;
C2(wnStart:Nn)=0;
C2Sum1(NnMidh)=d2wdZ20*dNn/4+d2wdZ2(NnMidh)*dNn/4;
C2(NnMidh)=(0*dNn/4 + C2Sum1(NnMidh)*dNn/4)/Z(NnMidh).^2;
for nn=NnMidh+1:Nn
        C2Sum1(nn)=C2Sum1(nn-1)+(d2wdZ2(nn-1)+d2wdZ2(nn))*dNn/2;
        C2(nn)=(C2Sum1(nn-1)*dNn/2+C2Sum1(nn)*dNn/2)./(Z(nn).^2);
end

C2Sum1(NnMidl)=d2wdZ20*dNn/4+d2wdZ2(NnMidl)*dNn/4;
C2(NnMidl)=(0*dNn/4 + C2Sum1(NnMidl)*dNn/4)/Z(NnMidl)^2;
for nn=NnMidl-1:-1:wnStart
        C2Sum1(nn)=C2Sum1(nn+1)+(d2wdZ2(nn+1)+d2wdZ2(nn))*dNn/2;
        C2(nn)=(C2Sum1(nn+1)*dNn/2+C2Sum1(nn)*dNn/2)./(Z(nn).^2);
end



for nn=NnMidh:Nn
    C3(nn)=((w(nn)-w0)-C1(nn).*Z(nn)-C2(nn).*(Z(nn)^2-1)-C20+3*C40)./(Z(nn).^3-3*Z(nn));
end
for nn=NnMidl:-1:wnStart
    C3(nn)=((w(nn)-w0)-C1(nn).*Z(nn)-C2(nn).*(Z(nn)^2-1)-C20+3*C40)./(Z(nn).^3-3*Z(nn));
end

w2(wnStart:Nn)=C0+C1(wnStart:Nn).*Z(wnStart:Nn)+C2(wnStart:Nn).*(Z(wnStart:Nn).^2-1)+C3(wnStart:Nn).*(Z(wnStart:Nn).^3-3*Z(wnStart:Nn));



%Mean is calculated and returned but it has nothing to do with calculation
%of hermite coefficients. It is for future use.
%--------------------------------------------------------------------
wMean=sum(w(wnStart:Nn).*Prob(wnStart:Nn));  %Calculate the mean. This is in general 

%nnMean=NnMidl;
%for nn=wnStart:Nn-1
%    if( (w(nn+1)>wMean) && (w(nn)<=wMean))
%        %ZMean=Z(nn)+(wMean-w(nn))/(w(nn+1)-w(nn))*(Z(nn+1)-Z(nn));
%        nnMean=nn;
%    end
%end

MeanFoundFlag=0;
nnMean=NnMidl;
for nn=NnMidl:Nn-1
    if( (w(nn+1)>wMean) && (w(nn)<=wMean)&& (MeanFoundFlag==0))
        %ZMean=Z(nn)+(wMean-w(nn))/(w(nn+1)-w(nn))*(Z(nn+1)-Z(nn));
        nnMean=nn;
        MeanFoundFlag=1;
    end
end

for nn=NnMidl-1:-1:wnStart
    if( (w(nn+1)>wMean) && (w(nn)<=wMean)&& (MeanFoundFlag==0))
        %ZMean=Z(nn)+(wMean-w(nn))/(w(nn+1)-w(nn))*(Z(nn+1)-Z(nn));
        nnMean=nn;
        MeanFoundFlag=1;
    end
end



 if(abs(wMean-w(nnMean))<abs(wMean-w(nnMean+1)))
 
 ZMean= InterpolateOrderN(5,wMean,w(nnMean-2),w(nnMean-1),w(nnMean),w(nnMean+1),w(nnMean+2),Z(nnMean-2),Z(nnMean-1),Z(nnMean),Z(nnMean+1),Z(nnMean+2));
 CSum=InterpolateOrderN(5,wMean,w(nnMean-2),w(nnMean-1),w(nnMean),w(nnMean+1),w(nnMean+2),CSum(nnMean-2),CSum(nnMean-1),CSum(nnMean),CSum(nnMean+1),CSum(nnMean+2));
 %C1=InterpolateOrderN(5,wMean,w(nnMean-2),w(nnMean-1),w(nnMean),w(nnMean+1),w(nnMean+2),C1(nnMean-2),C1(nnMean-1),C1(nnMean),C1(nnMean+1),C1(nnMean+2));
 Cm2=InterpolateOrderN(5,wMean,w(nnMean-2),w(nnMean-1),w(nnMean),w(nnMean+1),w(nnMean+2),C2(nnMean-2),C2(nnMean-1),C2(nnMean),C2(nnMean+1),C2(nnMean+2));
 end
 
 if(abs(wMean-w(nnMean))>=abs(wMean-w(nnMean+1)))
 
 ZMean= InterpolateOrderN(5,wMean,w(nnMean-1),w(nnMean),w(nnMean+1),w(nnMean+2),w(nnMean+3),Z(nnMean-1),Z(nnMean),Z(nnMean+1),Z(nnMean+2),Z(nnMean+3));
 CSum=InterpolateOrderN(5,wMean,w(nnMean-1),w(nnMean),w(nnMean+1),w(nnMean+2),w(nnMean+3),CSum(nnMean-1),CSum(nnMean),CSum(nnMean+1),CSum(nnMean+2),CSum(nnMean+3));
 %Cm1=InterpolateOrderN(5,wMean,w(nnMean-1),w(nnMean),w(nnMean+1),w(nnMean+2),w(nnMean+3),C1(nnMean-1),C1(nnMean),C1(nnMean+1),C1(nnMean+2),C1(nnMean+3));
 Cm2=InterpolateOrderN(5,wMean,w(nnMean-1),w(nnMean),w(nnMean+1),w(nnMean+2),w(nnMean+3),C2(nnMean-1),C2(nnMean),C2(nnMean+1),C2(nnMean+2),C2(nnMean+3));
 end
%ZMean=InterpolateOrderN(4,wMean,w(nnMean-1),w(nnMean),w(nnMean+1),w(nnMean+2),0,Z(nnMean-1),Z(nnMean),Z(nnMean+1),Z(nnMean+2),0);
%CSum=InterpolateOrderN(4,wMean,w(nnMean-1),w(nnMean),w(nnMean+1),w(nnMean+2),0,CSum(nnMean-1),CSum(nnMean),CSum(nnMean+1),CSum(nnMean+2),0);


Cm1=CSum./ZMean;


% if ((ZMean>=0)&&(ZMean<dNn/4))
%     Cm1=dwdZ0;
% end
% if ((ZMean>dNn/4)&&(ZMean<dNn/2))
%     Cm1=(dwdZ0*dNn/4+(ZMean-dNn/4)*dwdZ(NnMidh))/ZMean;
% end
% 
% if(ZMean>Z(NnMidh))
%     if(ZMean<Z(nnMean)+dNn/2)
%         Cm1=(C1(nnMean).*Z(nnMean)+dwdZ(nnMean).*(ZMean-Z(nnMean)))./ZMean;
%     end
%     if(ZMean>Z(nnMean)+dNn/2)
%         Cm1=(C1(nnMean).*Z(nnMean)+dwdZ(nnMean).*dNn/2+dwdZ(nnMean+1).*(ZMean-Z(nnMean)-dNn/2))./ZMean;
%     end
% end
% 
% if ((ZMean<0)&&(ZMean>-dNn/4))
%     Cm1=dwdZ0;
% end
% if ((ZMean<-dNn/4)&&(ZMean>-dNn/2))
%     Cm1=(dwdZ0*-dNn/4+(ZMean+dNn/4)*dwdZ(NnMidl))/ZMean;
% end
% 
% if(ZMean<Z(NnMidl))
%     if(ZMean>Z(nnMean+1)-dNn/2)
%         Cm1=(C1(nnMean+1).*Z(nnMean+1)+dwdZ(nnMean).*(ZMean-Z(nnMean+1)))./ZMean;
%     end
%     if(ZMean<Z(nnMean)-dNn/2)
%         Cm1=(C1(nnMean+1).*Z(nnMean+1)+dwdZ(nnMean+1).*-dNn/2+dwdZ(nnMean).*(ZMean-Z(nnMean+1)+dNn/2))./ZMean;
%     end
% end

    
Cm3=((wMean-w0)-Cm1.*ZMean-Cm2.*(ZMean.^2-1)-C20+3*C40)./(ZMean.^3-3*ZMean);






%Below we plot a graphical comparison of the original SDE variable w and
%the variable w1 reproduced from the coefficients of the hermite
%polynomials
% plot(Z(wnStart:Nn),w(wnStart:Nn),'g',Z(wnStart:Nn),w2(wnStart:Nn),'r')
%Sigma00

C1
C2
C0
% str=input('Look at the graphical comparison of w and w1 as a function of standard gaussian Z');
end

You will need the following function.

function [dwdZ,d2wdZ2,d3wdZ3,d4wdZ4] = First4Derivatives2ndOrderEqSpaced(wnStart,Nn,dNn,w,Z)

 d4wdZ4(wnStart)=(3*w(wnStart)-14*w(wnStart+1)+26*w(wnStart+2)-24*w(wnStart+3)+11*w(wnStart+4)-2*w(wnStart+5))/(dNn.^4);
 d4wdZ4(wnStart+1)=(2*w(wnStart)-9*w(wnStart+1)+16*w(wnStart+2)-14*w(wnStart+3)+6*w(wnStart+4)-1*w(wnStart+5))/(dNn.^4);
 d4wdZ4(wnStart+2:Nn-2)=(1*w(wnStart:Nn-4)-4*w(wnStart+1:Nn-3)+6*w(wnStart+2:Nn-2)-4*w(wnStart+3:Nn-1)+1*w(wnStart+4:Nn))/(dNn.^4);
 d4wdZ4(Nn-1)=(2*w(Nn)-9*w(Nn-1)+16*w(Nn-2)-14*w(Nn-3)+6*w(Nn-4)-1*w(Nn-5))/(dNn.^4);
 d4wdZ4(Nn)=(3*w(Nn)-14*w(Nn-1)+26*w(Nn-2)-24*w(Nn-3)+11*w(Nn-4)-2*w(Nn-5))/(dNn.^4);
 
 d3wdZ3(wnStart)=(-5*w(wnStart)+18*w(wnStart+1)-24*w(wnStart+2)+14*w(wnStart+3)-3*w(wnStart+4))/(2*dNn.^3);
 d3wdZ3(wnStart+1)=(-3*w(wnStart)+10*w(wnStart+1)-12*w(wnStart+2)+6*w(wnStart+3)-1*w(wnStart+4))/(2*dNn.^3);
 d3wdZ3(wnStart+2:Nn-2)=(-1*w(wnStart:Nn-4)+2*w(wnStart+1:Nn-3)+0*w(wnStart+2:Nn-2)-2*w(wnStart+3:Nn-1)+1*w(wnStart+4:Nn))/(2*dNn.^3);
 d3wdZ3(Nn-1)=(3*w(Nn)-10*w(Nn-1)+12*w(Nn-2)-6*w(Nn-3)+1*w(Nn-4))/(2*dNn.^3);
 d3wdZ3(Nn)=(5*w(Nn)-18*w(Nn-1)+24*w(Nn-2)-14*w(Nn-3)+3*w(Nn-4))/(2*dNn.^3);
 
 d2wdZ2(wnStart)=(2*w(wnStart)-5*w(wnStart+1)+4*w(wnStart+2)-1*w(wnStart+3))/(dNn.^2);
 d2wdZ2(wnStart+1:Nn-1)=(w(wnStart:Nn-2)-2*w(wnStart+1:Nn-1)+1*w(wnStart+2:Nn))/(dNn.^2);
 d2wdZ2(Nn)=(2*w(Nn)-5*w(Nn-1)+4*w(Nn-2)-1*w(Nn-3))/(dNn.^2);
 
 dwdZ(wnStart)=(-3*w(wnStart)+4*w(wnStart+1)-1*w(wnStart+2))/(2*dNn);
 dwdZ(wnStart+1:Nn-1)=(-1*w(wnStart:Nn-2)+1*w(wnStart+2:Nn))/(2*dNn);
 dwdZ(Nn)=(3*w(Nn)-4*w(Nn-1)+1*w(Nn-2))/(2*dNn);
 
end

Amin · March 14th, 2020, 5:54 pm

I have completely re-worked the structure of the program. I have removed looping from monte carlo and written all terms in a serial fashion both for 2nd order and fourth order monte carlo. I have also re-worked the part that calculates the drift, its derivatives and volatility in terms of hermite polynomials. With the new structure, you can easily know about the contribution to drift and volatility from a certain derivative as the indexing is based on derivatives of drift and volatility. Here is the new program

function [TrueMean,ItoHermiteMean,MCMean] = ItoHermiteMethodWilmott08Q40(x0,theta,kappa,gamma,sigma0,T)

%Copyright Ahsan Amin. Infiniti derivatives Technologies.
%Please fell free to connect on linkedin: linkedin.com/in/ahsan-amin-0a53334 
%or skype ahsan.amin2999
%In this program, I am simulating the SDE given as
%dy(t)=mu1 x(t)^beta1 dt + mu2 x(t)^beta2 dt +sigma x(t)^gamma dz(t)

%I have not directly simulated the SDE but simulated the transformed 
%Besse1l process version of the SDE and then changed coordinates to retreive
%the SDE in original coo
%rdinates.
%The present program will analytically evolve only the Bessel Process version of the
%SDE in transformed coordinates.
%In this version, the mean correction has been disabled. If you are
%simulating standard mean-reverting SV type SDEs, please enable the mean 
%correction by uncommenting the appropriate line in the body of code below.

dt=.125/16;   % Simulation time interval.%Fodiffusions close to zero
             %decrease dt for accuracy.
%T=5;
Tt=128*T;%16*16;     % Number of simulation levels. Terminal time= Tt*dt; //.125/32*32*16=2 year; 
OrderA=4;  %
OrderM=4;  %
%dtM=.125/8;%Monte carlo time interval size dtM.
%TtM=8*8;%Monte carlo number of simulation intervals.
dtM=dt*2;
TtM=Tt/2;


dNn=.2/1;   % Normal density subdivisions width. would change with number of subdivisions
Nn=48;  % No of normal density subdivisions
NnMidl=25;%One half density Subdivision left from mid of normal density(low)
NnMidh=26;%One half density subdivision right from the mid of normal density(high)
NnMid=4.0;

%theta=mu1/(-mu2);
%kappa=-mu2;

%x0=.25;   % starting value of SDE
beta1=0.0;
beta2=1.0;   % Second drift term power.
%gamma=.95;%50;   % volatility power.                                                                                                                                                                                                                                                                     
%kappa=.5;   %mean reversion parameter.
%theta=.075;%mean reversion target

%you can specify any general mu1 and mu2 and beta1 and beta2.
%The present program is not perfect but quite reasonably good.
%if there is only one term in drift with beta=0, there will be minor
%errors. This case is yet not covered.

mu1=+1*theta*kappa;   %first drift coefficient.
mu2=-1*kappa;    % Second drift coefficient.
%sigma0=1.0;%Volatility value
alpha=1;% x^alpha is being expanded. This is currently for monte carlo only.
alpha1=1-gamma;
                
w(1:Nn)=x0^(1-gamma)/(1-gamma);

Z(1:Nn)=(((1:Nn)-5.5)*dNn-NnMid);

Z
str=input('Look at Z');
ZProb(1)=normcdf(.5*Z(1)+.5*Z(2),0,1)-normcdf(.5*Z(1)+.5*Z(2)-dNn,0,1);
ZProb(Nn)=normcdf(.5*Z(Nn)+.5*Z(Nn-1)+dNn,0,1)-normcdf(.5*Z(Nn)+.5*Z(Nn-1),0,1);

for nn=2:Nn-1
    ZProb(nn)=normcdf(.5*Z(nn)+.5*Z(nn+1),0,1)-normcdf(.5*Z(nn)+.5*Z(nn-1),0,1);
    %Above calculate probability mass in each probability subdivision.
end

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

sigma11(1:OrderA+1)=0;
mu11(1:OrderA+1)=0;
mu22(1:OrderA+1)=0;
sigma22(1:OrderA+1)=0;
% index 1 correponds to zero level since matlab indexing starts at one. 
sigma11(1)=1;
mu11(1)=1;
mu22(1)=1;
sigma22(1)=1;

for k=1:(OrderA+1)
    if sigma0~=0
        sigma11(k)=sigma0^(k-1);
    end
    if mu1 ~= 0
        mu11(k)=mu1^(k-1);
    end
    if mu2 ~= 0
        mu22(k)=mu2^(k-1);
    end
    if sigma0~=0
        sigma22(k)=sigma0^(2*(k-1));
    end
end
%Ft(1:TtM+1,1:(OrderA+1),1:(OrderA+1),1:(OrderA+1),1:(OrderA+1))=0; %General time powers on hermite polynomials
Fp(1:(OrderA+1),1:(OrderA+1),1:(OrderA+1),1:(OrderA+1))=0;%General x powers on coefficients of hermite polynomials.
Fp1(1:(OrderA+1),1:(OrderA+1),1:(OrderA+1),1:(OrderA+1))=0;%General x powers for bessel transformed coordinates.

%YCoeff0 and YCoeff are coefficents for original coordinates monte carlo.
%YqCoeff0 and YqCoeff are bessel/lamperti version monte carlo.

YCoeff0(1:(OrderA+1),1:(OrderA+1),1:(OrderA+1),1:(OrderA+1))=0;
YqCoeff0(1:(OrderA+1),1:(OrderA+1),1:(OrderA+1),1:(OrderA+1))=0;
%Pre-compute the time and power exponent values in small multi-dimensional arrays
YCoeff = ItoTaylorCoeffsNew(alpha,beta1,beta2,gamma); %expand y^alpha where alpha=1;
YqCoeff = ItoTaylorCoeffsNew(alpha1,beta1,beta2,gamma);%expand y^alpha1 where alpha1=(1-gamma)
YqCoeff=YqCoeff/(1-gamma); %Transformed coordinates coefficients have to be 
%further divided by (1-gamma)

for k = 0 : (OrderA)
    for m = 0:k
        l4 = k - m + 1;
        for n = 0 : m
            l3 = m - n + 1;
            for j = 0:n
                l2 = n - j + 1;
                l1 = j + 1;
                %Ft(l1,l2,l3,l4) = dtM^((l1-1) + (l2-1) + (l3-1) + .5* (l4-1));
                Fp(l1,l2,l3,l4) = (alpha + (l1-1) * beta1 + (l2-1) * beta2 + (l3-1) * 2* gamma + (l4-1) * gamma ...
                    - (l1-1) - (l2-1) - 2* (l3-1) - (l4-1));
                Fp1(l1,l2,l3,l4) = (alpha1 + (l1-1) * beta1 + (l2-1) * beta2 + (l3-1) * 2* gamma + (l4-1) * gamma ...
                    - (l1-1) - (l2-1) - 2* (l3-1) - (l4-1));
                
                YCoeff0(l1,l2,l3,l4) =YCoeff(l1,l2,l3,l4).*mu11(l1).*mu22(l2).*sigma22(l3).*sigma11(l4);
                YqCoeff0(l1,l2,l3,l4) =YqCoeff(l1,l2,l3,l4).*mu11(l1).*mu22(l2).*sigma22(l3).*sigma11(l4);
                
            end
        end
    end
end

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
wnStart=1;%
d2wdZ2(1:Nn)=0;
dwdZ(1:Nn)=0;
yy(wnStart:Nn)=((1-gamma)*w(wnStart:Nn)).^(1/(1-gamma));
tic

for tt=1:Tt
  
     yy(wnStart:Nn)=((1-gamma)*w(wnStart:Nn)).^(1/(1-gamma));
    %[wMu0dt,dwMu0dtdw,d2wMu0dtdw2,c1,c2] = CalculateDriftAndDerivatives(w,wnStart,Nn,mu1,mu2,beta1,beta2,sigma0,gamma,dt);
    [wMu0dt,dwMu0dtdw,d2wMu0dtdw2,c1,c2,c3,c4] = CalculateDriftAndVolOrder4(w,wnStart,Nn,YqCoeff0,Fp1,gamma,dt)

    
    dw(wnStart:Nn)=c1(wnStart:Nn).*Z(wnStart:Nn) ;% ...
    dw2(wnStart:Nn)=dw(wnStart:Nn).^2;
    d2w(wnStart:Nn)=c2(wnStart:Nn).*(Z(wnStart:Nn).^2-1);
    d3w(wnStart:Nn)=c3(wnStart:Nn).*(Z(wnStart:Nn).^3-3*Z(wnStart:Nn));
    w(isnan(w)==1)=0;
    wMu0dt(isnan(wMu0dt)==1)=0;  
    wMeanPrev=sum(ZProb(wnStart:Nn).*w(wnStart:Nn));%Calculate the mean.
   
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%


[w0,Sigma00,Sigma11,Sigma22,Sigma33,Sigma1,Sigma2,Sigma3,Sigma0] = CalculateHermiteRepresentationOfDensityOrderThreeNew(w,Z,wnStart,Nn,dNn,NnMidh,NnMidl);
d2wdZ2(wnStart:Nn)=d2wdZ2(wnStart:Nn)+ d2wMu0dtdw2(wnStart:Nn).*dwdZ(wnStart:Nn).^2+dwMu0dtdw(wnStart:Nn).*d2wdZ2(wnStart:Nn);%+ ...
dwdZ(wnStart:Nn)=(Sigma00)+Sigma1(wnStart:Nn).*2.*Z(wnStart:Nn);

%dwdZN(wnStart+1:Nn-1)=(w(wnStart+2:Nn)-w(wnStart:Nn-2))/(2*dNn);
%dwdZN(wnStart)= InterpolateOrderN(4,Z(wnStart),Z(wnStart+1),Z(wnStart+2),Z(wnStart+3),Z(wnStart+4),0,dwdZN(wnStart+1),dwdZN(wnStart+2),dwdZN(wnStart+3),dwdZN(wnStart+4),0);
    
H2(wnStart:Nn)=(Z(wnStart:Nn).^2-1);      
CTerm(wnStart:Nn)=(d2wMu0dtdw2(wnStart:Nn).*dwdZ(wnStart:Nn).^2 - ...
          1*dwMu0dtdw(wnStart:Nn).*d2wdZ2(wnStart:Nn));       
Correction1(wnStart:Nn)=-sqrt(3)/2/(22/7)^2.*CTerm(wnStart:Nn).*H2(wnStart:Nn);      
 
if(tt==1)
w(wnStart:Nn)=wMeanPrev+wMu0dt(wnStart:Nn)+ ...
          sign(w(wnStart:Nn)-wMeanPrev+dw(wnStart:Nn)).* ...
          sqrt(abs(sign(w(wnStart:Nn)-wMeanPrev).*(w(wnStart:Nn)-wMeanPrev).^2+ ...
          sign(dw(wnStart:Nn)).*dw2(wnStart:Nn)));
end
if(tt>1)
    
    
     [C0,C1,C2,C3,w0,C20,wMean,ZMean,nnMean,Cm1,Cm2,Cm3] = CalculateHermitesOrderThree(w,Z,ZProb,wnStart,Nn,dNn,NnMidh,NnMidl);
    
     %Below is hermite decomposition of w
     %w(wnStart:Nn)=C0+C1(wnStart:Nn).*Z(wnStart:Nn)+C2(wnStart:Nn).*(Z(wnStart:Nn).^2-1);
     %Below is hermite decomposition of wMean with associated ZMean.
     %wMean=C0+Cm1.*ZMean+Cm2.*(ZMean^2-1);
     
     %[mC0,mC1,mC2,mw0,mC20] = CalculateHermitesOrderTwoWithoutMean(wMu0dt,Z,ZProb,wnStart,Nn,dNn,NnMidh,NnMidl);
     [mC0,mC1,mC2,mC3,Mu0,mC20,MuMean] = CalculateHermitesOrderThreeWithoutMean(wMu0dt,Z,ZProb,wnStart,Nn,dNn,NnMidh,NnMidl)
     %Below is hermite decomposition of drift   
     %wMu0dt(wnStart:Nn)=mC0+mC1(wnStart:Nn).*Z(wnStart:Nn)+mC2(wnStart:Nn).*(Z(wnStart:Nn).^2-1);
    %Correction1(wnStart:Nn)=-1/sqrt(1)/(22/7)^1.*mC2(wnStart:Nn).*H2(wnStart:Nn);      
     
 w(wnStart:Nn)=wMeanPrev+mC0+mC1(wnStart:Nn).*Z(wnStart:Nn)+mC2(wnStart:Nn).*(Z(wnStart:Nn).^2-1)+mC3(wnStart:Nn).*(Z(wnStart:Nn).^3-3*Z(wnStart:Nn))+ ...
         sign(C1(wnStart:Nn).*Z(wnStart:Nn)-Cm1.*ZMean+dw(wnStart:Nn)).* ...
         sqrt(abs(sign(C1(wnStart:Nn).*Z(wnStart:Nn)-Cm1.*ZMean).*(C1(wnStart:Nn).*Z(wnStart:Nn)-Cm1.*ZMean).^2+ ...
         sign(dw(wnStart:Nn)).*dw2(wnStart:Nn)))+ ...
         sign(C2(wnStart:Nn).*(Z(wnStart:Nn).^2-1)-Cm2.*(ZMean.^2-1)+d2w(wnStart:Nn)).* ...
         sqrt(abs(sign(C2(wnStart:Nn).*(Z(wnStart:Nn).^2-1)-Cm2.*(ZMean.^2-1)).*(C2(wnStart:Nn).*(Z(wnStart:Nn).^2-1)-Cm2.*(ZMean.^2-1)).^2+ ...
         sign(d2w(wnStart:Nn)).*d2w(wnStart:Nn).^2)) + ...
         sign(C3(wnStart:Nn).*(Z(wnStart:Nn).^3-3*Z(wnStart:Nn))-Cm3.*(ZMean.^3-3*ZMean)+d3w(wnStart:Nn)).* ...
         sqrt(abs((C3(wnStart:Nn).*(Z(wnStart:Nn).^3-3*Z(wnStart:Nn))-Cm3.*(ZMean.^3-3*ZMean)).^2+ ...
         sign(d3w(wnStart:Nn)).*d3w(wnStart:Nn).^2))+ ...
         Correction1(wnStart:Nn);       

end

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%Below are instability correction equations. There is instability in the 
% SDE representation equation roughly between -.5 SD to +.5 SD in normal terms.
%Instability correction Algorithm takes two ends that are here given as
%lower end=w(NnMidl-2) and upper end=w(NnMidh+3).
%In the block below the interpolation starts from lower end.
%1st equation is
%wll(NnMidl-1)=w(NnMidl-2)+(D1wl+CTheta*D2wl*dNn/1+.5*DeltaD1w)*dNn;
%Second equation is: wll(NnMidl)=wll(NnMidl-1)+(D1wl+CTheta*D2wl*dNn+1.5*DeltaD1w)*dNn;
%And so on.
%There is a sixth equation that is not written below(since it takes us to 
%the upper interpolation point which is already known) which is 
%wll(NnMidh+3)=wll(NnMidh+2)+(D1wl+CTheta*D2wl*dNn+5.5*DeltaD1w)*dNn;
%Sum of these six equations add up to the distance between upper and lower
%interpolation points.
%In the sum there are six instances of first derivative which is already 
%calculated at the boundary, six instances of second derivative which is also
%already calculated at lower boundary and 18.0 instances of DeltaD1w which is 
%unknown. This equation calculates the value of one unit of DeltaD1w addition
%DeltaD1w=(w(NnMidh+3)-w(NnMidl-2)-6*D1wl*dNn-6.0*CTheta*D2wl*dNn.^2.0/1.0)/18.0/dNn;
% I hope to find a better analytical method to tackle this issue. Friends
% are welcome to play with slightly changed related alternatives to the 
%above interpolation.
    CTheta=1.0;
    D1wl=(11/6*w(NnMidl-2)-3*w(NnMidl-3)+3/2*w(NnMidl-4)-1/3*w(NnMidl-5))/dNn;
    D2wl=(35/12*w(NnMidl-2) -26/3*w(NnMidl-3)+ 19/2*w(NnMidl-4)-14/3*w(NnMidl-5)+11/12*w(NnMidl-6))/dNn.^2;
    DeltaD1w=(w(NnMidh+3)-w(NnMidl-2)-6*D1wl*dNn-6.0*CTheta*D2wl*dNn.^2.0/1.0)/18.0/dNn;
    
    wll(NnMidl-1)=w(NnMidl-2)+  (D1wl+CTheta*D2wl*dNn+.5*DeltaD1w)*dNn;
    wll(NnMidl)=wll(NnMidl-1)+  (D1wl+CTheta*D2wl*dNn+1.5*DeltaD1w)*dNn;
    wll(NnMidh)=wll(NnMidl)+    (D1wl+CTheta*D2wl*dNn+2.5*DeltaD1w)*dNn;
    wll(NnMidh+1)=wll(NnMidh)+  (D1wl+CTheta*D2wl*dNn+3.5*DeltaD1w)*dNn;
    wll(NnMidh+2)=wll(NnMidh+1)+(D1wl+CTheta*D2wl*dNn+4.5*DeltaD1w)*dNn;
    %wll(NnMidh+3)=wll(NnMidh+2)+(D1wl+CTheta*D2wl*dNn+5.5*DeltaD1w)*dNn;
%The last equation is not used since point w(NnMidh+3)is given but it is used in 
%calculation of distance between two interpolation points. This distance is
%used to determine the value of DeltaD1w.

    D1wh=(-11/6*w(NnMidh+3)+3*w(NnMidh+4)-3/2*w(NnMidh+5)+1/3*w(NnMidh+6))/dNn;
    D2wh=(35/12*w(NnMidh+3) -26/3*w(NnMidh+4)+ 19/2*w(NnMidh+5)-14/3*w(NnMidh+6)+11/12*w(NnMidh+7))/dNn.^2;
    DeltaD1w=(w(NnMidh+3)-w(NnMidl-2)-6*D1wh*dNn-6.0*CTheta*D2wh*dNn.^2.0/1.0)/18.0/dNn;
%     DeltaD1w=(w(NnMidh+2)-w(NnMidl-1)-4*D1wh*dNn-16.0*D2wh*dNn.^2.0/1.0)/6.0/dNn;


    whh(NnMidh+2)=w(NnMidh+3)-(D1wh+CTheta*D2wh*dNn+.5*DeltaD1w)*dNn;
    whh(NnMidh+1)=w(NnMidh+2)-(D1wh+CTheta*D2wh*dNn+1.5* DeltaD1w)*dNn;
    whh(NnMidh)=whh(NnMidh+1)-(D1wh+CTheta*D2wh*dNn+2.5* DeltaD1w)*dNn;
    whh(NnMidl)=whh(NnMidh)-(D1wh+CTheta*D2wh*dNn+3.5* DeltaD1w)*dNn;
    whh(NnMidl-1)=whh(NnMidl)-(D1wh+CTheta*D2wh*dNn+4.5* DeltaD1w)*dNn;

    %whh(NnMidl-2)=whh(NnMidl-1)-(D1wh+CTheta*D2wh*dNn/1+5.5* DeltaD1w)*dNn;

% 
     w(NnMidh+2)=.5*whh(NnMidh+2)+.5*wll(NnMidh+2);
     w(NnMidh+1)=.5*whh(NnMidh+1)+.5*wll(NnMidh+1);
     w(NnMidh)=.5*whh(NnMidh)+.5*wll(NnMidh);
     w(NnMidl)=.5*whh(NnMidl)+.5*wll(NnMidl);
     w(NnMidl-1)=.5*whh(NnMidl-1)+.5*wll(NnMidl-1);



             
        w1(1:Nn-1)=w(1:Nn-1);
        w2(1:Nn-1)=w(2:Nn);
        w(w1(:)>w2(:))=0;%Be careful;might not universally hold;
     
        w(w<0)=0.0;
        for nn=1:Nn
            if(w(nn)<=0)
                wnStart=nn+1;
           end
        end
        
        %%1st order mean correction. We know the mean in original
        %%coordinates so I shift the density into original coordinates,
        %%apply the mean correction and then transform again into Lamperti
        %%coordinates. Algebra solves two equations in two unknowns.
        %%Equations are Sum_0^N{(Y_w(wnStart:Nn)-Y0)*W0}=1 and the second
        %%equation is Sum_0^N{Y_w(wnStart:Nn).*(Y_w(wnStart:Nn)-Y0)*W0}=u0
        %%Two unknows are Y0 and W0. u0 is known mean.
        tt;
        u0=theta+(x0-theta)*exp(-kappa*(tt*dt)); %analytic mean of the density
        
        %If you are not using stochastic volatility, replace above with
        %true mean otherwise results would become garbage. 

        Y_w(wnStart:Nn) = ((1-gamma)*w(wnStart:Nn)).^(1/(1-gamma));
    
        Yn2Pn=sum(Y_w(wnStart:Nn).*Y_w(wnStart:Nn).*ZProb(wnStart:Nn));
        YnPn=sum(Y_w(wnStart:Nn).*ZProb(wnStart:Nn));
        Pn=1.0;%%Sum(ZProb(1:Nn))
        Y0=(Yn2Pn-u0*YnPn)/(YnPn-u0*Pn);
    
        Y0Pn=Y0*Pn;
        W0=1/(YnPn-Y0Pn);
        YCorrected(wnStart:Nn)=Y_w(wnStart:Nn).*(Y_w(wnStart:Nn)-Y0)*W0;
        wCorrected(wnStart:Nn)=(YCorrected(wnStart:Nn).^(1-gamma))./(1-gamma);
   %     w(wnStart:Nn)=wCorrected(wnStart:Nn);
        
   %I have disabled the mean correction. The above mean correction is only valid for 
   %standard mean reverting stochastic volatility type SDEs. To enable mean
   %correction please uncomment the above last line in the block.
        
        
end

 y_w(1:Nn)=0;
 y_w(wnStart:Nn) = ((1-gamma)*w(wnStart:Nn)).^(1/(1-gamma));
 Dfy_w(wnStart:Nn)=0;
 Dfw(wnStart:Nn)=0;
for nn=wnStart+1:Nn-1
    Dfy_w(nn) = (y_w(nn + 1) - y_w(nn - 1))/(Z(nn + 1) - Z(nn - 1));
    Dfw(nn) = (w(nn + 1) - w(nn - 1))/(Z(nn + 1) - Z(nn - 1));
    %Change of variable derivative for densities
end
fy_w(1:Nn)=0;
fw(1:Nn)=0;
for nn = wnStart:Nn-1
    fy_w(nn) = (normpdf(Z(nn),0, 1))/abs(Dfy_w(nn));%Origianl coordinates density
    fw(nn) = (normpdf(Z(nn),0, 1))/abs(Dfw(nn));
end

toc
%str=input('Analytic Values calculated; Press a key to start monte carlo');

 rng(29079137, 'twister')
 paths=100000;
 YY(1:paths)=x0;  %Original process monte carlo.
 Random1(1:paths)=0;
 YYMean(1:TtM)=0;
 for tt=1:TtM
     Random1=randn(size(Random1));
     HermiteP1(1,1:paths)=1;
     HermiteP1(2,1:paths)=Random1(1:paths);
     HermiteP1(3,1:paths)=Random1(1:paths).^2-1;
     HermiteP1(4,1:paths)=Random1(1:paths).^3-3*Random1(1:paths);
     HermiteP1(5,1:paths)=Random1(1:paths).^4-6*Random1(1:paths).^2+3;

     YY(1:paths)=YY(1:paths) + ...
      (YCoeff0(1,1,2,1).*YY(1:paths).^Fp(1,1,2,1)+ ...
      YCoeff0(1,2,1,1).*YY(1:paths).^Fp(1,2,1,1)+ ...
      YCoeff0(2,1,1,1).*YY(1:paths).^Fp(2,1,1,1))*dtM + ...
      (YCoeff0(1,1,3,1).*YY(1:paths).^Fp(1,1,3,1)+ ...
      YCoeff0(1,2,2,1).*YY(1:paths).^Fp(1,2,2,1)+ ...
      YCoeff0(2,1,2,1).*YY(1:paths).^Fp(2,1,2,1)+ ...
      YCoeff0(1,3,1,1).*YY(1:paths).^Fp(1,3,1,1)+ ...
      YCoeff0(2,2,1,1).*YY(1:paths).^Fp(2,2,1,1)+ ...
      YCoeff0(3,1,1,1).*YY(1:paths).^Fp(3,1,1,1))*dtM^2 + ...
      ((YCoeff0(1,1,1,2).*YY(1:paths).^Fp(1,1,1,2).*sqrt(dtM))+ ...
    (YCoeff0(1,1,2,2).*YY(1:paths).^Fp(1,1,2,2)+ ...
    YCoeff0(1,2,1,2).*YY(1:paths).^Fp(1,2,1,2)+ ...
    YCoeff0(2,1,1,2).*YY(1:paths).^Fp(2,1,1,2)).*dtM^1.5) .*HermiteP1(2,1:paths) + ...
    ((YCoeff0(1,1,1,3).*YY(1:paths).^Fp(1,1,1,3) *dtM) + ...
    (YCoeff0(1,1,2,3).*YY(1:paths).^Fp(1,1,2,3)+ ...
    YCoeff0(1,2,1,3).*YY(1:paths).^Fp(1,2,1,3)+ ...
    YCoeff0(2,1,1,3).*YY(1:paths).^Fp(2,1,1,3)).*dtM^2).*HermiteP1(3,1:paths) + ...
    ((YCoeff0(1,1,1,4).*YY(1:paths).^Fp(1,1,1,4)*dtM^1.5 )).*HermiteP1(4,1:paths) + ...
    (YCoeff0(1,1,1,5).*YY(1:paths).^Fp(1,1,1,5)*dtM^2.0).*HermiteP1(5,1:paths);

%Uncomment for fourth order monte carlo

%   
%   YY(1:paths)=YY(1:paths) + ...
%       (YCoeff0(1,1,2,1).*YY(1:paths).^Fp(1,1,2,1)+YCoeff0(1,2,1,1).*YY(1:paths).^Fp(1,2,1,1)+ ...
%       YCoeff0(2,1,1,1).*YY(1:paths).^Fp(2,1,1,1))*dtM + ...
%       (YCoeff0(1,1,3,1).*YY(1:paths).^Fp(1,1,3,1)+YCoeff0(1,2,2,1).*YY(1:paths).^Fp(1,2,2,1)+ ...
%       YCoeff0(2,1,2,1).*YY(1:paths).^Fp(2,1,2,1)+YCoeff0(1,3,1,1).*YY(1:paths).^Fp(1,3,1,1)+ ...
%       YCoeff0(2,2,1,1).*YY(1:paths).^Fp(2,2,1,1)+YCoeff0(3,1,1,1).*YY(1:paths).^Fp(3,1,1,1))*dtM^2 + ...
%       (YCoeff0(1,1,4,1).*YY(1:paths).^Fp(1,1,4,1)+YCoeff0(1,2,3,1).*YY(1:paths).^Fp(1,2,3,1)+ ...
%       YCoeff0(2,1,3,1).*YY(1:paths).^Fp(2,1,3,1)+YCoeff0(1,3,2,1).*YY(1:paths).^Fp(1,3,2,1)+ ...
%       YCoeff0(2,2,2,1).*YY(1:paths).^Fp(2,2,2,1)+YCoeff0(3,1,2,1).*YY(1:paths).^Fp(3,1,2,1)+ ...
%       YCoeff0(1,4,1,1).*YY(1:paths).^Fp(1,4,1,1)+YCoeff0(2,3,1,1).*YY(1:paths).^Fp(2,3,1,1)+ ...
%       YCoeff0(3,2,1,1).*YY(1:paths).^Fp(3,2,1,1)+YCoeff0(4,1,1,1).*YY(1:paths).^Fp(4,1,1,1))*dtM^3 + ...
%        (YCoeff0(1,1,5,1).*YY(1:paths).^Fp(1,1,5,1)+YCoeff0(1,2,4,1).*yy(1:paths).^Fp(1,2,4,1)+ ...
%     YCoeff0(2,1,4,1).*YY(1:paths).^Fp(2,1,4,1)+YCoeff0(2,1,1,1).*yy(1:paths).^Fp(2,1,1,1)+ ...
%      YCoeff0(2,2,3,1).*YY(1:paths).^Fp(2,2,3,1)+ ...
%      YCoeff0(3,1,3,1).*YY(1:paths).^Fp(3,1,3,1)+YCoeff0(1,4,2,1).*YY(1:paths).^Fp(1,4,2,1)+ ...
%      YCoeff0(2,3,2,1).*YY(1:paths).^Fp(2,3,2,1)+YCoeff0(3,2,2,1).*YY(1:paths).^Fp(3,2,2,1)+ ...
%      YCoeff0(4,1,2,1).*YY(1:paths).^Fp(4,1,2,1)+YCoeff0(1,5,1,1).*YY(1:paths).^Fp(1,5,1,1)+ ...
%      YCoeff0(2,4,1,1).*YY(1:paths).^Fp(2,4,1,1)+ ...
%      YCoeff0(3,3,1,1).*YY(1:paths).^Fp(3,3,1,1)+YCoeff0(4,2,1,1).*YY(1:paths).^Fp(4,2,1,1)+ ...
%      YCoeff0(5,1,1,1).*YY(1:paths).^Fp(5,1,1,1))*dtM^4+ ...
%       ((YCoeff0(1,1,1,2).*YY(1:paths).^Fp(1,1,1,2).*sqrt(dtM))+ ...
%     (YCoeff0(1,1,2,2).*YY(1:paths).^Fp(1,1,2,2)+YCoeff0(1,2,1,2).*YY(1:paths).^Fp(1,2,1,2)+ ...
%     YCoeff0(2,1,1,2).*YY(1:paths).^Fp(2,1,1,2)).*dtM^1.5+ ...
%     (YCoeff0(1,1,3,2).*YY(1:paths).^Fp(1,1,3,2)+YCoeff0(1,2,2,2).*YY(1:paths).^Fp(1,2,2,2)+ ...
%     YCoeff0(2,1,2,2).*YY(1:paths).^Fp(2,1,2,2)+YCoeff0(1,3,1,2).*YY(1:paths).^Fp(1,3,1,2)+ ...
%     YCoeff0(2,2,1,2).*YY(1:paths).^Fp(2,2,1,2)+YCoeff0(3,1,1,2).*YY(1:paths).^Fp(3,1,1,2)).*dtM^2.5+ ...
%     (YCoeff0(1,1,4,2).*YY(1:paths).^Fp(1,1,4,2)+YCoeff0(1,2,3,2).*YY(1:paths).^Fp(1,2,3,2)+ ...
%     YCoeff0(2,1,3,2).*YY(1:paths).^Fp(2,1,3,2)+YCoeff0(1,3,2,2).*YY(1:paths).^Fp(1,3,2,2)+ ...
%     YCoeff0(2,2,2,2).*YY(1:paths).^Fp(2,2,2,2)+ YCoeff0(3,1,2,2).*YY(1:paths).^Fp(3,1,2,2)+ ...
%     YCoeff0(1,4,1,2).*YY(1:paths).^Fp(1,4,1,2)+YCoeff0(2,3,1,2).*YY(1:paths).^Fp(2,3,1,2)+ ...
%     YCoeff0(3,2,1,2).*YY(1:paths).^Fp(3,2,1,2)+YCoeff0(4,1,1,2).*YY(1:paths).^Fp(4,1,1,2)).*dtM^3.5) .*HermiteP1(2,1:paths) + ...
%     ((YCoeff0(1,1,1,3).*YY(1:paths).^Fp(1,1,1,3) *dtM) + ...
%     (YCoeff0(1,1,2,3).*YY(1:paths).^Fp(1,1,2,3)+YCoeff0(1,2,1,3).*YY(1:paths).^Fp(1,2,1,3)+ ...
%     YCoeff0(2,1,1,3).*YY(1:paths).^Fp(2,1,1,3)).*dtM^2+ ...
%     (YCoeff0(1,1,3,3).*YY(1:paths).^Fp(1,1,3,3)+YCoeff0(1,2,2,3).*YY(1:paths).^Fp(1,2,2,3)+ ...
%     YCoeff0(2,1,2,3).*YY(1:paths).^Fp(2,1,2,3) + YCoeff0(1,3,1,3).*YY(1:paths).^Fp(1,3,1,3)+ ...
%     YCoeff0(2,2,1,3).*YY(1:paths).^Fp(2,2,1,3)+YCoeff0(3,1,1,3).*YY(1:paths).^Fp(3,1,1,3)).*dtM^3).*HermiteP1(3,1:paths) + ...
%     ((YCoeff0(1,1,1,4).*YY(1:paths).^Fp(1,1,1,4)*dtM^1.5 )+ ...
%     (YCoeff0(1,1,2,4).*YY(1:paths).^Fp(1,1,2,4)+YCoeff0(1,2,1,4).*YY(1:paths).^Fp(1,2,1,4)+ ...
%     YCoeff0(2,1,1,4).*YY(1:paths).^Fp(2,1,1,4))*dtM^2.5).*HermiteP1(4,1:paths) + ...
%     (YCoeff0(1,1,1,5).*YY(1:paths).^Fp(1,1,1,5)*dtM^2.0).*HermiteP1(5,1:paths);
%     
 
     YYMean(tt)=YYMean(tt)+sum(YY(1:paths))/paths;
 end
 YY(YY<0)=0;
 disp('Original process average from monte carlo');
 MCMean=sum(YY(:))/paths %origianl coordinates monte carlo average.
 disp('Original process average from our simulation');
 ItoHermiteMean=sum(y_w(wnStart+1:Nn-1).*ZProb(wnStart+1:Nn-1)) %Original process average from coordinates 
 
 disp('true Mean only applicble to standard SV mean reverting type models otherwise disregard');
 TrueMean=theta+(x0-theta)*exp(-kappa*dt*Tt)%Mean reverting SDE original variable true average
 
 
 MaxCutOff=30;
 NoOfBins=round(500*gamma^2*1*sigma0);
 [YDensity,IndexOutY,IndexMaxY] = MakeDensityFromSimulation_Infiniti_NEW(YY,paths,NoOfBins,MaxCutOff );
 plot(y_w(wnStart+1:Nn-1),fy_w(wnStart+1:Nn-1),'r',IndexOutY(1:IndexMaxY),YDensity(1:IndexMaxY),'g');
 %plot(y_w(wnStart+1:Nn-1),fy_w(wnStart+1:Nn-1),'r',IndexOutY(1:IndexMaxY),YDensity(1:IndexMaxY),'g',Z(wnStart+1:Nn-1),fy_w(wnStart+1:Nn-1),'b');
 title('Graphical Comparison of Ito-Hermite Method and Monte Carlo Method Density')
 legend({'Ito-Hermite Density','Monte Carlo Density'},'Location','northeast')
 
 str=input('red line is density of SDE from Ito-Hermite method, green is monte carlo.');
 
end

Here is the new function for calculation of drift, its derivatives and volatility in terms of hermites. It is named order4 but it is actually order dt^3 upto four hermite polynomials.

function [wMu0dt,dwMu0dtdw,d2wMu0dtdw2,c1,c2,c3,c4] = CalculateDriftAndVolOrder4(w,wnStart,Nn,YqCoeff0,Fp1,gamma,dt)

yy(wnStart:Nn)=((1-gamma)*w(wnStart:Nn)).^(1/(1-gamma));

Fp2=Fp1/(1-gamma);

wMu0dt(wnStart:Nn)=(YqCoeff0(1,1,2,1).*yy(wnStart:Nn).^Fp1(1,1,2,1)+ ...
    YqCoeff0(1,2,1,1).*yy(wnStart:Nn).^Fp1(1,2,1,1)+ ...
    YqCoeff0(2,1,1,1).*yy(wnStart:Nn).^Fp1(2,1,1,1))*dt + ...
    (YqCoeff0(1,1,3,1).*yy(wnStart:Nn).^Fp1(1,1,3,1)+ ...
    YqCoeff0(1,2,2,1).*yy(wnStart:Nn).^Fp1(1,2,2,1)+ ...
    YqCoeff0(2,1,2,1).*yy(wnStart:Nn).^Fp1(2,1,2,1)+ ...
    YqCoeff0(1,3,1,1).*yy(wnStart:Nn).^Fp1(1,3,1,1)+ ...
    YqCoeff0(2,2,1,1).*yy(wnStart:Nn).^Fp1(2,2,1,1)+ ...
    YqCoeff0(3,1,1,1).*yy(wnStart:Nn).^Fp1(3,1,1,1))*dt^2 + ...
    (YqCoeff0(1,1,4,1).*yy(wnStart:Nn).^Fp1(1,1,4,1)+ ...
    YqCoeff0(1,2,3,1).*yy(wnStart:Nn).^Fp1(1,2,3,1)+ ...
    YqCoeff0(2,1,3,1).*yy(wnStart:Nn).^Fp1(2,1,3,1)+ ...
    YqCoeff0(1,3,2,1).*yy(wnStart:Nn).^Fp1(1,3,2,1)+ ...
    YqCoeff0(2,2,2,1).*yy(wnStart:Nn).^Fp1(2,2,2,1)+ ...
    YqCoeff0(3,1,2,1).*yy(wnStart:Nn).^Fp1(3,1,2,1)+ ...
    YqCoeff0(1,4,1,1).*yy(wnStart:Nn).^Fp1(1,4,1,1)+ ...
    YqCoeff0(2,3,1,1).*yy(wnStart:Nn).^Fp1(2,3,1,1)+ ...
    YqCoeff0(3,2,1,1).*yy(wnStart:Nn).^Fp1(3,2,1,1)+ ...
    YqCoeff0(4,1,1,1).*yy(wnStart:Nn).^Fp1(4,1,1,1))*dt^3;

dwMu0dtdw(wnStart:Nn)=(YqCoeff0(1,1,2,1).*Fp1(1,1,2,1).*((1-gamma).*w(wnStart:Nn)).^(-1+Fp2(1,1,2,1))+ ...
    YqCoeff0(1,2,1,1).*Fp1(1,2,1,1).*((1-gamma).*w(wnStart:Nn)).^(-1+Fp2(1,2,1,1))+ ...
    YqCoeff0(2,1,1,1).*Fp1(2,1,1,1).*((1-gamma).*w(wnStart:Nn)).^(-1+Fp2(2,1,1,1)))*dt + ...
    (YqCoeff0(1,1,3,1).*Fp1(1,1,3,1).*((1-gamma).*w(wnStart:Nn)).^(-1+Fp2(1,1,3,1))+ ...
    YqCoeff0(1,2,2,1).*Fp1(1,2,2,1).*((1-gamma).*w(wnStart:Nn)).^(-1+Fp2(1,2,2,1))+ ...
    YqCoeff0(2,1,2,1).*Fp1(2,1,2,1).*((1-gamma).*w(wnStart:Nn)).^(-1+Fp2(2,1,2,1))+ ...
    YqCoeff0(1,3,1,1).*Fp1(1,3,1,1).*((1-gamma).*w(wnStart:Nn)).^(-1+Fp2(1,3,1,1))+ ...
    YqCoeff0(2,2,1,1).*Fp1(2,2,1,1).*((1-gamma).*w(wnStart:Nn)).^(-1+Fp2(2,2,1,1))+ ...
    YqCoeff0(3,1,1,1).*Fp1(3,1,1,1).*((1-gamma).*w(wnStart:Nn)).^(-1+Fp2(3,1,1,1)))*dt^2 + ...
    (YqCoeff0(1,1,4,1).*Fp1(1,1,4,1).*((1-gamma).*w(wnStart:Nn)).^(-1+Fp2(1,1,4,1))+ ...
    YqCoeff0(1,2,3,1).*Fp1(1,2,3,1).*((1-gamma).*w(wnStart:Nn)).^(-1+Fp2(1,2,3,1))+ ...
    YqCoeff0(2,1,3,1).*Fp1(2,1,3,1).*((1-gamma).*w(wnStart:Nn)).^(-1+Fp2(2,1,3,1))+ ...
    YqCoeff0(1,3,2,1).*Fp1(1,3,2,1).*((1-gamma).*w(wnStart:Nn)).^(-1+Fp2(1,3,2,1))+ ...
    YqCoeff0(2,2,2,1).*Fp1(2,2,2,1).*((1-gamma).*w(wnStart:Nn)).^(-1+Fp2(2,2,2,1))+ ...
    YqCoeff0(3,1,2,1).*Fp1(3,1,2,1).*((1-gamma).*w(wnStart:Nn)).^(-1+Fp2(3,1,2,1))+ ...
    YqCoeff0(1,4,1,1).*Fp1(1,4,1,1).*((1-gamma).*w(wnStart:Nn)).^(-1+Fp2(1,4,1,1))+ ...
    YqCoeff0(2,3,1,1).*Fp1(2,3,1,1).*((1-gamma).*w(wnStart:Nn)).^(-1+Fp2(2,3,1,1))+ ...
    YqCoeff0(3,2,1,1).*Fp1(3,2,1,1).*((1-gamma).*w(wnStart:Nn)).^(-1+Fp2(3,2,1,1))+ ...
    YqCoeff0(4,1,1,1).*Fp1(4,1,1,1).*((1-gamma).*w(wnStart:Nn)).^(-1+Fp2(4,1,1,1)))*dt^3;

d2wMu0dtdw2(wnStart:Nn)=(1-gamma).*((YqCoeff0(1,1,2,1).*Fp1(1,1,2,1).*(-1+Fp2(1,1,2,1)).*((1-gamma).*w(wnStart:Nn)).^(-2+Fp2(1,1,2,1))+ ...
    YqCoeff0(1,2,1,1).*Fp1(1,2,1,1).*(-1+Fp2(1,2,1,1)).*((1-gamma).*w(wnStart:Nn)).^(-2+Fp2(1,2,1,1))+ ...
    YqCoeff0(2,1,1,1).*Fp1(2,1,1,1).*(-1+Fp2(2,1,1,1)).*((1-gamma).*w(wnStart:Nn)).^(-2+Fp2(2,1,1,1)))*dt + ...
    (YqCoeff0(1,1,3,1).*Fp1(1,1,3,1).*(-1+Fp2(1,1,3,1)).*((1-gamma).*w(wnStart:Nn)).^(-2+Fp2(1,1,3,1))+ ...
    YqCoeff0(1,2,2,1).*Fp1(1,2,2,1).*(-1+Fp2(1,2,2,1)).*((1-gamma).*w(wnStart:Nn)).^(-2+Fp2(1,2,2,1))+ ...
    YqCoeff0(2,1,2,1).*Fp1(2,1,2,1).*(-1+Fp2(2,1,2,1)).*((1-gamma).*w(wnStart:Nn)).^(-2+Fp2(2,1,2,1))+ ...
    YqCoeff0(1,3,1,1).*Fp1(1,3,1,1).*(-1+Fp2(1,3,1,1)).*((1-gamma).*w(wnStart:Nn)).^(-2+Fp2(1,3,1,1))+ ...
    YqCoeff0(2,2,1,1).*Fp1(2,2,1,1).*(-1+Fp2(2,2,1,1)).*((1-gamma).*w(wnStart:Nn)).^(-2+Fp2(2,2,1,1))+ ...
    YqCoeff0(3,1,1,1).*Fp1(3,1,1,1).*(-1+Fp2(3,1,1,1)).*((1-gamma).*w(wnStart:Nn)).^(-2+Fp2(3,1,1,1)))*dt^2 + ...
    (YqCoeff0(1,1,4,1).*Fp1(1,1,4,1).*(-1+Fp2(1,1,4,1)).*((1-gamma).*w(wnStart:Nn)).^(-2+Fp2(1,1,4,1))+ ...
    YqCoeff0(1,2,3,1).*Fp1(1,2,3,1).*(-1+Fp2(1,2,3,1)).*((1-gamma).*w(wnStart:Nn)).^(-2+Fp2(1,2,3,1))+ ...
    YqCoeff0(2,1,3,1).*Fp1(2,1,3,1).*(-1+Fp2(2,1,3,1)).*((1-gamma).*w(wnStart:Nn)).^(-2+Fp2(2,1,3,1))+ ...
    YqCoeff0(1,3,2,1).*Fp1(1,3,2,1).*(-1+Fp2(1,3,2,1)).*((1-gamma).*w(wnStart:Nn)).^(-2+Fp2(1,3,2,1))+ ...
    YqCoeff0(2,2,2,1).*Fp1(2,2,2,1).*(-1+Fp2(2,2,2,1)).*((1-gamma).*w(wnStart:Nn)).^(-2+Fp2(2,2,2,1))+ ...
    YqCoeff0(3,1,2,1).*Fp1(3,1,2,1).*(-1+Fp2(3,1,2,1)).*((1-gamma).*w(wnStart:Nn)).^(-2+Fp2(3,1,2,1))+ ...
    YqCoeff0(1,4,1,1).*Fp1(1,4,1,1).*(-1+Fp2(1,4,1,1)).*((1-gamma).*w(wnStart:Nn)).^(-2+Fp2(1,4,1,1))+ ...
    YqCoeff0(2,3,1,1).*Fp1(2,3,1,1).*(-1+Fp2(2,3,1,1)).*((1-gamma).*w(wnStart:Nn)).^(-2+Fp2(2,3,1,1))+ ...
    YqCoeff0(3,2,1,1).*Fp1(3,2,1,1).*(-1+Fp2(3,2,1,1)).*((1-gamma).*w(wnStart:Nn)).^(-2+Fp2(3,2,1,1))+ ...
    YqCoeff0(4,1,1,1).*Fp1(4,1,1,1).*(-1+Fp2(4,1,1,1)).*((1-gamma).*w(wnStart:Nn)).^(-2+Fp2(4,1,1,1)))*dt^3);


c1(wnStart:Nn)=YqCoeff0(1,1,1,2).*yy(wnStart:Nn).^Fp1(1,1,1,2).*sqrt(dt)+ ...
    (YqCoeff0(1,1,2,2).*yy(wnStart:Nn).^Fp1(1,1,2,2)+YqCoeff0(1,2,1,2).*yy(wnStart:Nn).^Fp1(1,2,1,2)+ ...
    YqCoeff0(2,1,1,2).*yy(wnStart:Nn).^Fp1(2,1,1,2)).*dt^1.5+ ...
    (YqCoeff0(1,1,3,2).*yy(wnStart:Nn).^Fp1(1,1,3,2)+YqCoeff0(1,2,2,2).*yy(wnStart:Nn).^Fp1(1,2,2,2)+ ...
    YqCoeff0(2,1,2,2).*yy(wnStart:Nn).^Fp1(2,1,2,2)+YqCoeff0(1,3,1,2).*yy(wnStart:Nn).^Fp1(1,3,1,2)+ ...
    YqCoeff0(2,2,1,2).*yy(wnStart:Nn).^Fp1(2,2,1,2)+YqCoeff0(3,1,1,2).*yy(wnStart:Nn).^Fp1(3,1,1,2)).*dt^2.5;


c2(wnStart:Nn)=YqCoeff0(1,1,1,3).*yy(wnStart:Nn).^Fp1(1,1,1,3) *dt + ...
    (YqCoeff0(1,1,2,3).*yy(wnStart:Nn).^Fp1(1,1,2,3)+YqCoeff0(1,2,1,3).*yy(wnStart:Nn).^Fp1(1,2,1,3)+ ...
    YqCoeff0(2,1,1,3).*yy(wnStart:Nn).^Fp1(2,1,1,3)).*dt^2+ ...
    (YqCoeff0(1,1,3,3).*yy(wnStart:Nn).^Fp1(1,1,3,3)+YqCoeff0(1,2,2,3).*yy(wnStart:Nn).^Fp1(1,2,2,3)+ ...
    YqCoeff0(2,1,2,3).*yy(wnStart:Nn).^Fp1(2,1,2,3) + YqCoeff0(1,3,1,3).*yy(wnStart:Nn).^Fp1(1,3,1,3)+ ...
    YqCoeff0(2,2,1,3).*yy(wnStart:Nn).^Fp1(2,2,1,3)+YqCoeff0(3,1,1,3).*yy(wnStart:Nn).^Fp1(3,1,1,3)).*dt^3;

c3(wnStart:Nn)=YqCoeff0(1,1,1,4).*yy(wnStart:Nn).^Fp1(1,1,1,4)*dt^1.5 + ...
    (YqCoeff0(1,1,2,4).*yy(wnStart:Nn).^Fp1(1,1,2,4)+YqCoeff0(1,2,1,4).*yy(wnStart:Nn).^Fp1(1,2,1,4)+ ...
    YqCoeff0(2,1,1,4).*yy(wnStart:Nn).^Fp1(2,1,1,4))*dt^2.5;

c4(wnStart:Nn)=YqCoeff0(1,1,1,5).*yy(wnStart:Nn).^Fp1(1,1,1,5)*dt^2.0;


end

Here is another re-worked function that you will need.

function [w0,Sigma00,Sigma11,Sigma22,Sigma33,Sigma1,Sigma2,Sigma3,Sigma0] = CalculateHermiteRepresentationOfDensityOrderThreeNew(w,Z,wnStart,Nn,dNn,NnMidh,NnMidl)

%This program find hermite representation of the SDE variable w to 
%tjird Order as a function of three hermite polynomials.
%This program is not perfectly precise and has very small errors so you
%will have to be careful using it everywhere.
%Here wnStart is the starting grid point of the SDE vector
%mostly wnStart=1.
%Please note we are representing SDE Variable w as a function of hermite 
%polynomials. It is not the density which can however be found by change of variable 
%derivative of w with respect to Z using standard gaussian variable.
%Nn is the index of last density grid point.
%So w(wnStart:Nn) represents the SDE density variable grid.
%  Z(wnStart:Nn) represents the associated value of standard Gaussian on
%  the grid. w and Z can be associated by matching appropriate CDF points
%  on the grid.
%I suppose the grid is uniform in Z and dNn is the spacing between and 
% Z(n) and Z(n+1) two points on the grid.


Z0=0;
w0= InterpolateOrderN6(6,Z0,Z(NnMidh+2),Z(NnMidh+1),Z(NnMidh),Z(NnMidl),Z(NnMidl-1),Z(NnMidl-2),w(NnMidh+2),w(NnMidh+1),w(NnMidh),w(NnMidl),w(NnMidl-1),w(NnMidl-2));


%Below Calculate the vol associated with each point on the grid
for nn=wnStart:Nn
    Sigma0(nn)=(w(nn)-w0)./(Z(nn)-Z0);  
 end

%Below calculate the associated Sigma00 which is a single volatility associated with
%the first orthogonal hermite polynomial.

Sigma00= InterpolateOrderN(4,Z0,Z(NnMidh+1),Z(NnMidh),Z(NnMidl),Z(NnMidl-1),0,Sigma0(NnMidh+1),Sigma0(NnMidh),Sigma0(NnMidl),Sigma0(NnMidl-1),0);


%Below calculate the associated first derivative of Sigma0 with respect to
%Z on the grid.
dSigma0dZ(wnStart)=(-3*Sigma0(wnStart)+4*Sigma0(wnStart+1)-1*Sigma0(wnStart+2))/(2*dNn);
dSigma0dZ(wnStart+1:Nn-1)=(-1*Sigma0(wnStart:Nn-2)+1*Sigma0(wnStart+2:Nn))/(2*dNn);
dSigma0dZ(Nn)=(3*Sigma0(Nn)-4*Sigma0(Nn-1)+1*Sigma0(Nn-2))/(2*dNn);
dSigma0dZ(wnStart+2:Nn-2)= (1*Sigma0(wnStart:Nn-4)-8*Sigma0(wnStart+1:Nn-3)+0*Sigma0(wnStart+2:Nn-2)+8*Sigma0(wnStart+3:Nn-1)-1*Sigma0(wnStart+4:Nn))/(12*dNn);
dSigma0dZ0 = (-9*Sigma0(NnMidl-2)+125*Sigma0(NnMidl-1)-2250*Sigma0(NnMidl)+2250*Sigma0(NnMidh)-125*Sigma0(NnMidh+1)+9*Sigma0(NnMidh+2))/(3840*(.5*dNn));


%Following is the operation related to equation(4) in post 860.     
SSigma1(NnMidh)=(.5*dSigma0dZ0+.5*dSigma0dZ(NnMidh))*dNn/2;
for nn=NnMidh+1:Nn
        SSigma1(nn)=SSigma1(nn-1)+(.5*dSigma0dZ(nn-1)+.5*dSigma0dZ(nn))*dNn;
end

SSigma1(NnMidl)=-(.5*dSigma0dZ0+.5*dSigma0dZ(NnMidl))*dNn/2;
for nn=NnMidl-1:-1:wnStart
        SSigma1(nn)=SSigma1(nn+1)-(.5*dSigma0dZ(nn+1)+.5*dSigma0dZ(nn))*dNn;
end

Sigma1(wnStart:Nn)=SSigma1(wnStart:Nn)./Z(wnStart:Nn);

Sigma11= InterpolateOrderN6(6,Z0,Z(NnMidh+2),Z(NnMidh+1),Z(NnMidh),Z(NnMidl),Z(NnMidl-1),Z(NnMidl-2),Sigma1(NnMidh+2),Sigma1(NnMidh+1),Sigma1(NnMidh),Sigma1(NnMidl),Sigma1(NnMidl-1),Sigma1(NnMidl-2));

%Now below we represent the SDE variable w in terms of first two hermite[/font][/size]
%polynomials and this associated value of w produced by hermite polynomials
%coefficients is called w1.
   
% w1(wnStart:Nn)=wMean-Sigma00*ZMean-Sigma1(wnStart:Nn).*(ZMean.^2-1)+ ...
% Sigma00*Z(wnStart:Nn)+ ...
% Sigma1(wnStart:Nn).*(Z(wnStart:Nn).^2-1);

w0(wnStart:Nn)=w0+Sigma0(wnStart:Nn).*Z(wnStart:Nn);


w1=w0;
w2=w0;
%w1(wnStart:Nn)=w0+ Sigma00*Z(wnStart:Nn)+ ...
% Sigma1(wnStart:Nn).*(Z(wnStart:Nn).^2-1);

%w2(wnStart:Nn)=w0+Sigma00*Z(wnStart:Nn)+ ...
% Sigma1(wnStart:Nn).*(Z(wnStart:Nn).^2-1);


dSigma1dZ(wnStart)=(-3*Sigma1(wnStart)+4*Sigma1(wnStart+1)-1*Sigma1(wnStart+2))/(2*dNn);
dSigma1dZ(wnStart+1:Nn-1)=(-1*Sigma1(wnStart:Nn-2)+1*Sigma1(wnStart+2:Nn))/(2*dNn);
dSigma1dZ(Nn)=(3*Sigma1(Nn)-4*Sigma1(Nn-1)+1*Sigma1(Nn-2))/(2*dNn);

dSigma1dZ0 = (-9*Sigma1(NnMidl-2)+125*Sigma1(NnMidl-1)-2250*Sigma1(NnMidl)+2250*Sigma1(NnMidh)-125*Sigma1(NnMidh+1)+9*Sigma1(NnMidh+2))/(3840*(.5*dNn));

SSigma2(NnMidh)=(.5*dSigma1dZ0+.5*dSigma1dZ(NnMidh))*dNn/2;
for nn=NnMidh+1:Nn
        SSigma2(nn)=SSigma2(nn-1)+(.5*dSigma1dZ(nn-1)+.5*dSigma1dZ(nn))*dNn;
end
SSigma2(NnMidl)=-(.5*dSigma1dZ0+.5*dSigma1dZ(NnMidl))*dNn/2;
for nn=NnMidl-1:-1:wnStart
        SSigma2(nn)=SSigma2(nn+1)-(.5*dSigma1dZ(nn+1)+.5*dSigma1dZ(nn))*dNn;
end

Sigma2(wnStart:Nn)=SSigma2(wnStart:Nn)./Z(wnStart:Nn);

Sigma22= InterpolateOrderN6(6,Z0,Z(NnMidh+2),Z(NnMidh+1),Z(NnMidh),Z(NnMidl),Z(NnMidl-1),Z(NnMidl-2),Sigma2(NnMidh+2),Sigma2(NnMidh+1),Sigma2(NnMidh),Sigma2(NnMidl),Sigma2(NnMidl-1),Sigma2(NnMidl-2));

dSigma2dZ(wnStart)=(-3*Sigma2(wnStart)+4*Sigma2(wnStart+1)-1*Sigma2(wnStart+2))/(2*dNn);
dSigma2dZ(wnStart+1:Nn-1)=(-1*Sigma2(wnStart:Nn-2)+1*Sigma2(wnStart+2:Nn))/(2*dNn);
dSigma2dZ(Nn)=(3*Sigma2(Nn)-4*Sigma2(Nn-1)+1*Sigma2(Nn-2))/(2*dNn);

dSigma2dZ0 = (-9*Sigma2(NnMidl-2)+125*Sigma2(NnMidl-1)-2250*Sigma2(NnMidl)+2250*Sigma2(NnMidh)-125*Sigma2(NnMidh+1)+9*Sigma2(NnMidh+2))/(3840*(.5*dNn));

SSigma3(NnMidh)=(.5*dSigma2dZ0+.5*dSigma2dZ(NnMidh))*dNn/2;
for nn=NnMidh+1:Nn
        SSigma3(nn)=SSigma3(nn-1)+(.5*dSigma2dZ(nn-1)+.5*dSigma2dZ(nn))*dNn;
end
SSigma3(NnMidl)=-(.5*dSigma2dZ0+.5*dSigma2dZ(NnMidl))*dNn/2;
for nn=NnMidl-1:-1:wnStart
        SSigma3(nn)=SSigma3(nn+1)-(.5*dSigma2dZ(nn+1)+.5*dSigma2dZ(nn))*dNn;
end

Sigma3(wnStart:Nn)=SSigma3(wnStart:Nn)./Z(wnStart:Nn);

Sigma33= InterpolateOrderN6(6,Z0,Z(NnMidh+2),Z(NnMidh+1),Z(NnMidh),Z(NnMidl),Z(NnMidl-1),Z(NnMidl-2),Sigma3(NnMidh+2),Sigma3(NnMidh+1),Sigma3(NnMidh),Sigma3(NnMidl),Sigma3(NnMidl-1),Sigma3(NnMidl-2));

end

Here is another function

function [C0,C1,C2,C3,w0,C20,wMean,ZMean,nnMean,Cm1,Cm2,Cm3] = CalculateHermitesOrderThree(w,Z,Prob,wnStart,Nn,dNn,NnMidh,NnMidl)

%-------------------------------------------------------------------------
%Below hermite coefficients calculation starts. InterpolateOrderN is a
%program that interpolates the values of w0 at Z=0 since in our set up grid
%points are located at Z(NnMidl)=-.1 and Z(NnMidh)=.1 and there are no grid
%points at Z=0.

Z0=0;
%w0= InterpolateOrderN(4,Z0,Z(NnMidh),Z(NnMidh+1),Z(NnMidl),Z(NnMidl-1),0,w(NnMidh),w(NnMidh+1),w(NnMidl),w(NnMidl-1),0);
w0= InterpolateOrderN6(6,Z0,Z(NnMidh+2),Z(NnMidh+1),Z(NnMidh),Z(NnMidl),Z(NnMidl-1),Z(NnMidl-2),w(NnMidh+2),w(NnMidh+1),w(NnMidh),w(NnMidl),w(NnMidl-1),w(NnMidl-2));
%dwdZ0=(w(NnMidh)-w(NnMidl))./dNn;
%dwdZ0= (1*w(NnMidl-1)-27*w(NnMidl)+27*w(NnMidh)-1*w(NnMidh+1))/(48*(.5*dNn));
dwdZ0 = (-9*w(NnMidl-2)+125*w(NnMidl-1)-2250*w(NnMidl)+2250*w(NnMidh)-125*w(NnMidh+1)+9*w(NnMidh+2))/(3840*(.5*dNn));
%d2wdZ20 = (-1*w(NnMidl-1)+81*w(NnMidl)-160*w0+81*w(NnMidh)-1*w(NnMidh+1))/(72*(.5*dNn)^2);
%d2wdZ20 = (-1*w(NnMidl-1)+81*w(NnMidl)-160*w0+81*w(NnMidh)-1*w(NnMidh+1))/(72*(.5*dNn)^2)
d2wdZ20 = (-5*w(NnMidl-2)+39*w(NnMidl-1)-34*w(NnMidl)-34*w(NnMidh)+39*w(NnMidh+1)-5*w(NnMidh+2))/(192*(.5*dNn)^2);


%d4wdZ40 = (-3*w(NnMidl-2)+65*w(NnMidl-1)-510*w(NnMidl)+896*w0-510*w(NnMidh)+65*w(NnMidh+1)-3*w(NnMidh+2))/(240*(.5*dNn)^2)

d4wdZ40 = (1*w(NnMidl-2)-3*w(NnMidl-1)+2*w(NnMidl)+2*w(NnMidh)-3*w(NnMidh+1)+1*w(NnMidh+2))/(32*(.5*dNn)^2);

%str=input('Look at derivatives at zero');



%Calculate first hermite polynomial coefficients C1 below.
dwdZ(wnStart)=(-3*w(wnStart)+4*w(wnStart+1)-1*w(wnStart+2))/(2*dNn);
dwdZ(wnStart+1:Nn-1)=(-1*w(wnStart:Nn-2)+1*w(wnStart+2:Nn))/(2*dNn);
dwdZ(Nn)=(3*w(Nn)-4*w(Nn-1)+1*w(Nn-2))/(2*dNn);
dwdZ(wnStart+2:Nn-2)= (1*w(wnStart:Nn-4)-8*w(wnStart+1:Nn-3)+0*w(wnStart+2:Nn-2)+8*w(wnStart+3:Nn-1)-1*w(wnStart+4:Nn))/(12*dNn);





CSum(wnStart:Nn)=0.0;
C1(wnStart:Nn)=0;
C1(NnMidh)=.5*dwdZ0+.5*dwdZ(NnMidh);
CSum(NnMidh)=C1(NnMidh).*Z(NnMidh);
for nn=NnMidh+1:Nn
        CSum(nn)=(C1(nn-1).*Z(nn-1)+(.5*dwdZ(nn-1)+.5*dwdZ(nn))*dNn);
        C1(nn)=CSum(nn)./Z(nn);
end
C1(NnMidl)=.5*dwdZ0+.5*dwdZ(NnMidl);
CSum(NnMidl)=C1(NnMidl).*Z(NnMidl);
for nn=NnMidl-1:-1:wnStart
        CSum(nn)=(C1(nn+1).*Z(nn+1)-(.5*dwdZ(nn+1)+.5*dwdZ(nn))*dNn);
        C1(nn)=CSum(nn)./Z(nn);
end

%Calculate second hermite coefficients below. Here C20 are coefficients of
%second hermite polynomial at Z=0 
C20=.5*d2wdZ20;
C40=1*1/24*d4wdZ40;
C0=w0+C20-3*C40;


[dwdZ,d2wdZ2,d3wdZ3,d4wdZ4] = First4Derivatives2ndOrderEqSpaced(wnStart,Nn,dNn,w,Z);
d2wdZ2(wnStart+2:Nn-2)= (-1*w(wnStart:Nn-4)+16*w(wnStart+1:Nn-3)-30*w(wnStart+2:Nn-2)+16*w(wnStart+3:Nn-1)-1*w(wnStart+4:Nn))/(12*dNn^2);


C2Sum1(wnStart:Nn)=0.0;
C2(wnStart:Nn)=0;
C2Sum1(NnMidh)=d2wdZ20*dNn/4+d2wdZ2(NnMidh)*dNn/4;
C2(NnMidh)=(0*dNn/4 + C2Sum1(NnMidh)*dNn/4)/Z(NnMidh).^2;
for nn=NnMidh+1:Nn
        C2Sum1(nn)=C2Sum1(nn-1)+(d2wdZ2(nn-1)+d2wdZ2(nn))*dNn/2;
        C2(nn)=(C2Sum1(nn-1)*dNn/2+C2Sum1(nn)*dNn/2)./(Z(nn).^2);
end

C2Sum1(NnMidl)=d2wdZ20*dNn/4+d2wdZ2(NnMidl)*dNn/4;
C2(NnMidl)=(0*dNn/4 + C2Sum1(NnMidl)*dNn/4)/Z(NnMidl)^2;
for nn=NnMidl-1:-1:wnStart
        C2Sum1(nn)=C2Sum1(nn+1)+(d2wdZ2(nn+1)+d2wdZ2(nn))*dNn/2;
        C2(nn)=(C2Sum1(nn+1)*dNn/2+C2Sum1(nn)*dNn/2)./(Z(nn).^2);
end



for nn=NnMidh:Nn
    C3(nn)=((w(nn)-w0)-C1(nn).*Z(nn)-C2(nn).*(Z(nn)^2-1)-C20+3*C40)./(Z(nn).^3-3*Z(nn));
end
for nn=NnMidl:-1:wnStart
    C3(nn)=((w(nn)-w0)-C1(nn).*Z(nn)-C2(nn).*(Z(nn)^2-1)-C20+3*C40)./(Z(nn).^3-3*Z(nn));
end

w2(wnStart:Nn)=C0+C1(wnStart:Nn).*Z(wnStart:Nn)+C2(wnStart:Nn).*(Z(wnStart:Nn).^2-1)+C3(wnStart:Nn).*(Z(wnStart:Nn).^3-3*Z(wnStart:Nn));



%Mean is calculated and returned but it has nothing to do with calculation
%of hermite coefficients. It is for future use.
%--------------------------------------------------------------------
wMean=sum(w(wnStart:Nn).*Prob(wnStart:Nn));  %Calculate the mean. This is in general 

%nnMean=NnMidl;
%for nn=wnStart:Nn-1
%    if( (w(nn+1)>wMean) && (w(nn)<=wMean))
%        %ZMean=Z(nn)+(wMean-w(nn))/(w(nn+1)-w(nn))*(Z(nn+1)-Z(nn));
%        nnMean=nn;
%    end
%end

MeanFoundFlag=0;
nnMean=NnMidl;
for nn=NnMidl:Nn-1
    if( (w(nn+1)>wMean) && (w(nn)<=wMean)&& (MeanFoundFlag==0))
        %ZMean=Z(nn)+(wMean-w(nn))/(w(nn+1)-w(nn))*(Z(nn+1)-Z(nn));
        nnMean=nn;
        MeanFoundFlag=1;
    end
end

for nn=NnMidl-1:-1:wnStart
    if( (w(nn+1)>wMean) && (w(nn)<=wMean)&& (MeanFoundFlag==0))
        %ZMean=Z(nn)+(wMean-w(nn))/(w(nn+1)-w(nn))*(Z(nn+1)-Z(nn));
        nnMean=nn;
        MeanFoundFlag=1;
    end
end



 if(abs(wMean-w(nnMean))<abs(wMean-w(nnMean+1)))
 
 ZMean= InterpolateOrderN(5,wMean,w(nnMean-2),w(nnMean-1),w(nnMean),w(nnMean+1),w(nnMean+2),Z(nnMean-2),Z(nnMean-1),Z(nnMean),Z(nnMean+1),Z(nnMean+2));
 CSum=InterpolateOrderN(5,wMean,w(nnMean-2),w(nnMean-1),w(nnMean),w(nnMean+1),w(nnMean+2),CSum(nnMean-2),CSum(nnMean-1),CSum(nnMean),CSum(nnMean+1),CSum(nnMean+2));
 %C1=InterpolateOrderN(5,wMean,w(nnMean-2),w(nnMean-1),w(nnMean),w(nnMean+1),w(nnMean+2),C1(nnMean-2),C1(nnMean-1),C1(nnMean),C1(nnMean+1),C1(nnMean+2));
 Cm2=InterpolateOrderN(5,wMean,w(nnMean-2),w(nnMean-1),w(nnMean),w(nnMean+1),w(nnMean+2),C2(nnMean-2),C2(nnMean-1),C2(nnMean),C2(nnMean+1),C2(nnMean+2));
 end
 
 if(abs(wMean-w(nnMean))>=abs(wMean-w(nnMean+1)))
 
 ZMean= InterpolateOrderN(5,wMean,w(nnMean-1),w(nnMean),w(nnMean+1),w(nnMean+2),w(nnMean+3),Z(nnMean-1),Z(nnMean),Z(nnMean+1),Z(nnMean+2),Z(nnMean+3));
 CSum=InterpolateOrderN(5,wMean,w(nnMean-1),w(nnMean),w(nnMean+1),w(nnMean+2),w(nnMean+3),CSum(nnMean-1),CSum(nnMean),CSum(nnMean+1),CSum(nnMean+2),CSum(nnMean+3));
 %Cm1=InterpolateOrderN(5,wMean,w(nnMean-1),w(nnMean),w(nnMean+1),w(nnMean+2),w(nnMean+3),C1(nnMean-1),C1(nnMean),C1(nnMean+1),C1(nnMean+2),C1(nnMean+3));
 Cm2=InterpolateOrderN(5,wMean,w(nnMean-1),w(nnMean),w(nnMean+1),w(nnMean+2),w(nnMean+3),C2(nnMean-1),C2(nnMean),C2(nnMean+1),C2(nnMean+2),C2(nnMean+3));
 end
%ZMean=InterpolateOrderN(4,wMean,w(nnMean-1),w(nnMean),w(nnMean+1),w(nnMean+2),0,Z(nnMean-1),Z(nnMean),Z(nnMean+1),Z(nnMean+2),0);
%CSum=InterpolateOrderN(4,wMean,w(nnMean-1),w(nnMean),w(nnMean+1),w(nnMean+2),0,CSum(nnMean-1),CSum(nnMean),CSum(nnMean+1),CSum(nnMean+2),0);


Cm1=CSum./ZMean;


% if ((ZMean>=0)&&(ZMean<dNn/4))
%     Cm1=dwdZ0;
% end
% if ((ZMean>dNn/4)&&(ZMean<dNn/2))
%     Cm1=(dwdZ0*dNn/4+(ZMean-dNn/4)*dwdZ(NnMidh))/ZMean;
% end
% 
% if(ZMean>Z(NnMidh))
%     if(ZMean<Z(nnMean)+dNn/2)
%         Cm1=(C1(nnMean).*Z(nnMean)+dwdZ(nnMean).*(ZMean-Z(nnMean)))./ZMean;
%     end
%     if(ZMean>Z(nnMean)+dNn/2)
%         Cm1=(C1(nnMean).*Z(nnMean)+dwdZ(nnMean).*dNn/2+dwdZ(nnMean+1).*(ZMean-Z(nnMean)-dNn/2))./ZMean;
%     end
% end
% 
% if ((ZMean<0)&&(ZMean>-dNn/4))
%     Cm1=dwdZ0;
% end
% if ((ZMean<-dNn/4)&&(ZMean>-dNn/2))
%     Cm1=(dwdZ0*-dNn/4+(ZMean+dNn/4)*dwdZ(NnMidl))/ZMean;
% end
% 
% if(ZMean<Z(NnMidl))
%     if(ZMean>Z(nnMean+1)-dNn/2)
%         Cm1=(C1(nnMean+1).*Z(nnMean+1)+dwdZ(nnMean).*(ZMean-Z(nnMean+1)))./ZMean;
%     end
%     if(ZMean<Z(nnMean)-dNn/2)
%         Cm1=(C1(nnMean+1).*Z(nnMean+1)+dwdZ(nnMean+1).*-dNn/2+dwdZ(nnMean).*(ZMean-Z(nnMean+1)+dNn/2))./ZMean;
%     end
% end

    
Cm3=((wMean-w0)-Cm1.*ZMean-Cm2.*(ZMean.^2-1)-C20+3*C40)./(ZMean.^3-3*ZMean);






%Below we plot a graphical comparison of the original SDE variable w and
%the variable w1 reproduced from the coefficients of the hermite
%polynomials
% plot(Z(wnStart:Nn),w(wnStart:Nn),'g',Z(wnStart:Nn),w2(wnStart:Nn),'r')
%Sigma00

C1
C2
C0
% str=input('Look at the graphical comparison of w and w1 as a function of standard gaussian Z');
end

Amin · March 16th, 2020, 12:39 pm

Since in my previous post, I have given a re-worked version of the code for higher order monte carlo simulations of SDEs, I will like to point out the relevant posts in this thread.
First of all, I started in this thread when I was working on analytics about SDEs and I discovered (on my own, though I am sure other mathematicians must have known about it) a special integral form of Taylor series which I called method of iterated integrals in which a dependent variable of (that is dependent on) one or more independent variables is expanded in terms of successive integrals of derivatives evaluated at an initial value of independent variables to get the value of the dependent variable in the neighborhood(in terms of independent variables) of the starting values of the independent variables. When I wrote it the expansion equation for the SDE in my notes, I knew that it will surely solve the SDEs to higher order and I started this thread. I still did not know all the details that were required to completely solve the problem.

Here is my post#37 on May 2nd 2016 where I first wrote the proper equations in a formal way on this forum.

Amin:

Cuch, here you could see why those integrals are helpful. I have tried to explain in simple terms.We take a general stochastic differential equation.[$]x(t)=x(0)+\int _0^t\mu (x(s)) \text{ds}+\int _0^t\sigma (x(s)) \text{dz}(s)[$] Eq(1)Where [$]\text{$\mu $(x(s))}[$] and [$]\text{ $\sigma $(x(s))}[$] are x dependent functions and are stochastic due to stochastic nature of x. It is unfortunate that when we use the above notation, we always substitute [$]\text{$\mu $(x(0))}[$] for [$]\text{$\mu $(x(s))}[$] and [$]\text{ $\sigma $(x(0))}[$] for [$]\text{ $\sigma $(x(s))}[$] as is standard in monte carlo and other numerical analysis and little effort has been made to understand the time dependent nature and evolution of [$]\text{$\mu $(x(s))}[$] and [$]\text{ $\sigma $(x(s))}[$] in between the simulation intervals. As I earlier said that \text{$\mu $(x(s))} and [$]\text{ $\sigma $(x(s))}[$] are functions that depend on x and are stochastic due to stochastic nature of x. We define the evolution of these functions as[$]\mu (x(s))=\mu (x(0))+\int _0^s \frac{\partial \mu (x(v))}{\partial x}\mu (x(v))\text{dv}+\int _0^s \frac{\partial \mu (x(v))}{\partial x}\sigma (x(v))\text{dz}(v)+.5\int _0^s \frac{\partial ^2\mu (x(v))}{\partial x^2}\sigma ^2(x(v)) \text{dv}[$] Eq(2)[$]\sigma (x(s))=\sigma (x(0))+\int _0^s \frac{\partial \sigma (x(v))}{\partial x}\mu (x(v))\text{dv}+\int _0^s \frac{\partial \sigma (x(v))}{\partial x}\sigma (x(v))\text{dz}(v)+.5\int _0^s \frac{\partial ^2\sigma (x(v))}{\partial x^2}\sigma ^2(x(v)) \text{dv}[$] Eq(3)We can easily make the observation that x dependent terms in the above integrals from 0 to s could be expanded by application of Ito formula into a time zero term and several other time dependent terms. We substitute equations 2 and 3 in Eq(1) to get[$]x(t)=x(0)+\int _0^t\left(\mu (x(0))+\int _0^s \frac{\partial \mu (x(v))}{\partial x}\mu (x(v))\text{dv}+\int _0^s \frac{\partial \mu (x(v))}{\partial x}\sigma (x(v))\text{dz}(v)+.5\int _0^s \frac{\partial ^2\mu (x(v))}{\partial x^2}.\sigma ^2(x(v)) \text{dv}\right) \text{ds}+[$][$]\int _0^t\left(\sigma (x(0))+\int _0^s \frac{\partial \sigma (x(v))}{\partial x}\mu (x(v))\text{dv}+\int _0^s \frac{\partial \sigma (x(v))}{\partial x}\sigma (x(v))\text{dz}(v)+.5\int _0^s \frac{\partial ^2\sigma (x(v))}{\partial x^2}\sigma ^2(x(v)) \text{dv}\right) \text{dz}(s)[$] Eq(4)Simplifying, we get[$]x(t)=x(0)+\int _0^t\mu (x(0))\text{ds}+\int _0^t\int _0^s \frac{\partial \mu (x(v))}{\partial x}\mu (x(v))\text{dvds}+\int _0^t\int _0^s \frac{\partial \mu (x(v))}{\partial x}\sigma (x(v))\text{dz}(v)\text{ds}+.5\int _0^t\int _0^s \frac{\partial ^2\mu (x(v))}{\partial x^2}\left.\sigma ^2(x(v)) \text{dv}\right) \text{ds}+\int _0^t\sigma (x(0))\text{dz}(s)+\int _0^t\int _0^s \frac{\partial \sigma (x(v))}{\partial x}\mu (x(v))\text{dvdz}(s)[$][$]+\int _0^t\int _0^s \frac{\partial \sigma (x(v))}{\partial x}\sigma (x(v))\text{dz}(v)\text{dz}(s)+.5\int _0^t\int _0^s \frac{\partial ^2\sigma (x(v))}{\partial x^2}\sigma ^2(x(v)) \text{dv}\text{dz}(s)[$] Eq(5) In the expression above only two single integrals [$]\int _0^t\mu (x(0))\text{ds}[$]and [$]\int _0^t\sigma (x(0))\text{dz}(s)[$] depend upon time zero value of x and the rest of the double integrals all depend upon forward time dependent values and have to be expanded again similarly and this process could continue until derivatives existed(though we would generally always stop at third or fourth level if we could atain desired precision). We expand just the first double integral in Eq(5) further as an example[$]\int _0^t\int _0^s \frac{\partial \mu (x(v))}{\partial x}\mu (x(v))\text{dvds}=\int _0^t\int _0^s \frac{\partial \mu (x(0))}{\partial x}\mu (x(0))\text{dvds}+\int _0^t\int _0^s\int _0^v\frac{\partial }{\partial x} \left[\frac{\partial \mu (x(u))}{\partial x}\mu (x(u))\right]\mu (x(u))\text{dudvds}[$][$]+\int _0^t\int _0^s\int _0^v\frac{\partial }{\partial x} \left[\frac{\partial \mu (x(u))}{\partial x}\mu (x(u))\right]\sigma (x(u))\text{dz}(u)\text{dvds}+\int _0^t\int _0^s\int _0^v.5\frac{\partial ^2}{\partial x^2} \left[\frac{\partial \mu (x(u))}{\partial x}\mu (x(u))\right]\sigma ^2(x(u))\text{dudvds}[$] Eq(6)We could continue further but if further derivatives continue to exist, we have to ultimately truncate the expansion at some level and we could simply substitute time zero values of drift and volatility for forward time dependent values of drift and volatility. If we decide to stop the expansion at second level, we could write from Eq(5) (Please notice that 2nd equality in equation below is approximate equality)[$]x(t)=x(0)+\int _0^t\mu (x(0))\text{ds}+\int _0^t\int _0^s \frac{\partial \mu (x(v))}{\partial x}\mu (x(v))\text{dvds}+\int _0^t\int _0^s \frac{\partial \mu (x(v))}{\partial x}\sigma (x(v))\text{dz}(v)\text{ds}+.5\int _0^t\int _0^s \frac{\partial ^2\mu (x(v))}{\partial x^2}\left.\sigma ^2(x(v)) \text{dv}\right) \text{ds}[$][$]+\int _0^t\sigma (x(0))\text{dz}(s)+\int _0^t\int _0^s \frac{\partial \sigma (x(v))}{\partial x}\mu (x(v))\text{dvdz}(s)+\int _0^t\int _0^s \frac{\partial \sigma (x(v))}{\partial x}\sigma (x(v))\text{dz}(v)\text{dz}(s)+.5\int _0^t\int _0^s \frac{\partial ^2\sigma (x(v))}{\partial x^2}\sigma ^2(x(v)) \text{dv}\text{dz}(s)[$][$]=x(0)+\int _0^t\mu (x(0))\text{ds}+\int _0^t\int _0^s \frac{\partial \mu (x(0))}{\partial x}\mu (x(0))\text{dvds}+\int _0^t\int _0^s \frac{\partial \mu (x(0))}{\partial x}\sigma (x(0))\text{dz}(v)\text{ds}+.5\int _0^t\int _0^s \frac{\partial ^2\mu (x(0))}{\partial x^2}\left.\sigma ^2(x(0)) \text{dv}\right) \text{ds}[$][$]+\int _0^t\sigma (x(0))\text{dz}(s)+\int _0^t\int _0^s \frac{\partial \sigma (x(0))}{\partial x}\mu (x(0))\text{dvdz}(s)+\int _0^t\int _0^s \frac{\partial \sigma (x(0))}{\partial x}\sigma (x(0))\text{dz}(v)\text{dz}(s)+.5\int _0^t\int _0^s \frac{\partial ^2\sigma (x(0))}{\partial x^2}\sigma ^2(x(0)) \text{dv}\text{dz}(s)[$][$]=x(0)+\mu (x(0))\int _0^t\text{ds}+\frac{\partial \mu (x(0))}{\partial x}\mu (x(0))\int _0^t\int _0^s \text{dvds}+\frac{\partial \mu (x(0))}{\partial x}\sigma (x(0))\int _0^t\int _0^s \text{dz}(v)\text{ds}+.5\frac{\partial ^2\mu (x(0))}{\partial x^2}\sigma ^2(x(0)) \int _0^t\int _0^s \text{dv} \text{ds}[$][$]+\sigma (x(0))\int _0^t\text{dz}(s)+\frac{\partial \sigma (x(0))}{\partial x}\mu (x(0))\int _0^t\int _0^s \text{dvdz}(s)+\frac{\partial \sigma (x(0))}{\partial x}\sigma (x(0))\int _0^t\int _0^s \text{dz}(v)\text{dz}(s)+.5\frac{\partial ^2\sigma (x(0))}{\partial x^2}\sigma ^2(x(0))\int _0^t\int _0^s\text{dv}\text{dz}(s)[$]In case we wanted to go to fourth or hififth or even higher level, we could have continued the repeated expansion and finally replaced forward values at their time zero values but If we only wanted to continue to third level, we could have truncated the further series and replaced forward values by their time zero values and could have included integrals of the kind from Eq(6). Please recall that when we wrote Eq(6), we expanded just one integral and other integral terms have to be similarly expanded.(Please notice that 2nd equality in equation below is approximate equality)[$]\int _0^t\int _0^s \frac{\partial \mu (x(v))}{\partial x}\mu (x(v))\text{dvds}[$][$]=\int _0^t\int _0^s \frac{\partial \mu (x(0))}{\partial x}\mu (x(0))\text{dvds}+\int _0^t\int _0^s\int _0^v\frac{\partial }{\partial x} \left[\frac{\partial \mu (x(u))}{\partial x}\mu (x(u))\right]\mu (x(u))\text{dudvds}+\int _0^t\int _0^s\int _0^v\frac{\partial }{\partial x} \left[\frac{\partial \mu (x(u))}{\partial x}\mu (x(u))\right]\sigma (x(u))\text{dz}(u)\text{dvds}+\int _0^t\int _0^s\int _0^v.5\frac{\partial ^2}{\partial x^2} \left[\frac{\partial \mu (x(u))}{\partial x}\mu (x(u))\right]\sigma ^2(x(u))\text{dudvds}[$][$]=\frac{\partial \mu (x(0))}{\partial x}\mu (x(0))\int _0^t\int _0^s \text{dvds}+\frac{\partial }{\partial x} \left[\frac{\partial \mu (x(0))}{\partial x}\mu (x(0))\right]\mu (x(0))\int _0^t\int _0^s\int _0^v\text{dudvds}+\frac{\partial }{\partial x} \left[\frac{\partial \mu (x(0))}{\partial x}\mu (x(0))\right]\sigma (x(0))\int _0^t\int _0^s\int _0^v\text{dz}(u)\text{dvds}+.5\frac{\partial ^2}{\partial x^2} \left[\frac{\partial \mu (x(0))}{\partial x}\mu (x(0))\right]\sigma ^2(x(0))\int _0^t\int _0^s\int _0^v\text{dudvds}[$]Cuch, here is the answer to your question in an earlier post why these integrals are helpful. As you can see in the above equation, once we have taken initial time zero values of x dependent coefficients, the integrals to be evaluated are of the kind [$]\int _0^t\int _0^s\int _0^v\text{dudvds}[$],[$]\int _0^t\int _0^s\int _0^v\text{dz}(u)\text{dvds}[$], [$]\int _0^t\int _0^s\int _0^v\text{dz}(u)\text{dz}(v)\text{ds}[$] and [$]\int _0^t\int _0^s\int _0^v\text{dz}(u)\text{dz}(v)\text{dz}(s)[$] which I mentioned how to evaluate in a previous post using ito isometry and simple hermite polynomials.

.
.
I was working to develop this into perfectly concrete formulas for simulation but I was forcefully hospitalized on totally flimsy excuses because our army on behest of foreign agencies forced psychiatrists to declare me mentally ill and detain me. It really obstructed my research and I was not able to work properly for a while. After I was released from detention of several weeks, I decided to first solve the problem of solution of ODEs as it was relatively easier and I knew my method was equally applicable to ODEs. Here is the link to paper I completed on ODEs. https://papers.ssrn.com/sol3/papers.cfm?abstract_id=2872598
I wanted to post the paper on arXiV and was able to secure sponsorship. On my account, I first noticed that status of the paper was "submitted" but then status changed to rejected after two days. Here is the abstract of the ODEs paper.

Abstract

Our goal is to give a very simple, effective and intuitive algorithm for the solution of initial value problem of ODEs of 1st and arbitrary higher order with general i.e. constant, variable or nonlinear coefficients and the systems of these ordinary differential equations. We find an expansion of the differential equation/function to get an infinite series containing iterated integrals evaluated solely at initial values of the dependent variables in the ordinary differential equation. Our series represents the true series expansion of the closed form solution of the ordinary differential equation. The method can also be used easily for general 2nd and higher order ordinary differential equations. We explain with examples the steps to solution of initial value problems of 1st order ordinary differential equations and later follow with more examples for linear and non-linear 2nd order and third order ODEs. We have given mathematica code for the solution of nth order ordinary differential equations and show with examples how to use the general code on 1st, 2nd and third order ordinary differential equations. We also give mathematica code for the solution of systems of a large number of general ordinary differential equations each of arbitrary order. We give an example showing the ease with which our method calculates the solution of system of three non-linear second order ordinary differential equations.

Here is a relevant post#208 written on Tue Dec 20, 2016 2:23 pm that explains in simple terms the ideas behind the application of the method to ODEs and SDEs.

Another question: OP started with SDE, now it's a discussion on ODE. What's the link?
The method I have proposed is a general way to solve integrals that involve functions of dependent variables and one(possibly more) independent variable where the dependence structure is given as dictated by the ODE or SDE itself. Solution of ordinary differential equations can be written as a deterministic integral while the solution of stochastic differential equations can be written as a stochastic integral. Both deterministic and stochastic integrals can be solved by the method of iterated integrals. We also know that solution to drift of an SDE is given by an ODE which is solved just like we solved the ODEs in my paper(Most are very simple). In the solution to stochastic integrals, we have iterated integrals some/all of which can be stochastic or deterministic or a mix of both. The method originated while I was working on SDEs hence I started out with the current name of the thread. Obviously solution to drift of an SDE is given by solution to an ODE.

After I completed the above paper, I was again detained in a mental hospital despite that I was very perfectly healthy because bribes were given to people in authority in Pakistan and psychiatrists would declare me mentally ill despite that I was thoroughly composed, perfectly sane and never even spoke to anyone in a louder voice . After I was released from several weeks of detention, I started doing my work again and slowly resumed my research.
Here is nother post#208 written on Thu Nov 23, 2017 10:48 am where I explained the idea again

Amin:

Suppose we want to solve the following general SDE

[$]\text{dX} (\tau) = \mu (X(\tau)) \text{d$\tau $}+ \sigma(X(\tau)) \text{dz}( \tau) [$] Equation(1)

The formal solution of the above SDE is given as
[$]X(t)=X(t_0)+\int_{t_0}^t \mu (X(\tau)) \, d\tau+\int _{t_0}^t\sigma(X(\tau)) dz(\tau )[$] Equation(2)

We want to solve both integrand terms in above equation as
[$] \mu (X(\tau))=\mu (X(t_0))+\int_{t_0}^\tau d\mu (X(s))[$] Equation(3)
and
[$] \sigma (X(\tau))=\sigma (X(t_0))+\int_{t_0}^\tau d\sigma (X(s))[$] Equation(4)
When we substitute above equation(3) and equation(4) in equation(2), we get

[$]X(t)=X(t_0)+\int_{t_0}^t \mu (X(t_0)) \, d\tau+\int_{t_0}^t \int_{t_0}^\tau d\mu (X(s)) d\tau [$]
[$]+\int _{t_0}^t\sigma(X(t_0)) dz(\tau )+\int _{t_0}^t \int_{t_0}^\tau d\sigma (X(s)) dz(\tau )[$] Equation(5)

From Ito expansion of [$]d\mu (X(s)) [$] and [$]d\sigma (X(s)) [$], we know that
[$]d\mu (X(s))=\frac{\partial }{\partial X}[\mu ( X(s))]\mu (X(s)) \text{ds}+\frac{\partial }{\partial X}[\mu (X(s))]\sigma (X(s)) \text{dz(s)}[$]
[$]+0.5 \frac{\partial ^2[\mu ( X(s))]}{\partial X^2} \sigma ( X(s))^2\text{ds}[$] Equation(6)

and
[$]d\sigma ( X(s))=\frac{\partial }{\partial X}[\sigma ( X(s))]\mu (X(s)) \text{ds}+\frac{\partial }{\partial X}[\sigma (X(s))]\sigma (X(s)) \text{dz(s)}[$]
[$]+0.5 \frac{\partial ^2[\sigma ( X(s))]}{\partial X^2} \sigma ( X(s))^2\text{ds}[$] Equation(7)

We substitute equation (6) and Equation(7) in equation(5)

[$]X(t)=X(t_0)+\int_{t_0}^t \mu (X(t_0)) \, d\tau[$]
[$]+\int_{t_0}^t \int_{t_0}^\tau \frac{\partial }{\partial X}[\mu ( X(s))]\mu (X(s)) \text{ds} d\tau+\int_{t_0}^t \int_{t_0}^\tau \frac{\partial }{\partial X}[\mu (X(s))]\sigma (X(s)) \text{dz(s)} d\tau[$]
[$]+\int_{t_0}^t \int_{t_0}^\tau 0.5 \frac{\partial ^2[\mu ( X(s))]}{\partial X^2} \sigma ( X(s))^2\text{ds} d\tau[$]
[$]+\int _{t_0}^t\sigma(X(t_0)) dz(\tau )[$]
[$]+\int _{t_0}^t \int_{t_0}^\tau \frac{\partial }{\partial X}[\sigma ( X(s))]\mu (X(s)) \text{ds} dz(\tau )[$]
[$]+\int _{t_0}^t \int_{t_0}^\tau \frac{\partial }{\partial X}[\sigma ( X(s))]\sigma (X(s)) dz(s ) dz(\tau )[$]
[$]+\int_{t_0}^t \int_{t_0}^\tau 0.5 \frac{\partial ^2[\sigma ( X(s))]}{\partial X^2} \sigma ( X(s))^2\text{ds} dz(\tau )[$] Equation(8)
In a further order expansion, we evaluate terms at time 's' at [$]t_0[$] and add a third order integral. I am not showing the third order integral but up to second order, we will have

[$]X(t)=X(t_0)+\int_{t_0}^t \mu (X(t_0)) \, d\tau[$]
[$]+\int_{t_0}^t \int_{t_0}^\tau \frac{\partial }{\partial X}[\mu ( X(t_0))]\mu (X(t_0)) \text{ds} d\tau+\int_{t_0}^t \int_{t_0}^\tau \frac{\partial }{\partial X}[\mu (X(t_0))]\sigma (X(t_0)) \text{dz(s)} d\tau[$]
[$]+\int_{t_0}^t \int_{t_0}^\tau 0.5 \frac{\partial ^2[\mu ( X(t_0))]}{\partial X^2} \sigma ( X(t_0))^2\text{ds} d\tau[$]
[$]+\int _{t_0}^t\sigma(X(t_0)) dz(\tau )[$]
[$]+\int _{t_0}^t \int_{t_0}^\tau \frac{\partial }{\partial X}[\sigma ( X(t_0))]\mu (X(t_0)) \text{ds} dz(\tau )[$]
[$]+\int _{t_0}^t \int_{t_0}^\tau \frac{\partial }{\partial X}[\sigma ( X(t_0))]\sigma (X(t_0)) dz(s ) dz(\tau )[$]
[$]+\int_{t_0}^t \int_{t_0}^\tau 0.5 \frac{\partial ^2[\sigma ( X(t_0))]}{\partial X^2} \sigma ( X(t_0))^2\text{ds} dz(\tau )[$] Equation(9)

Since all the terms are evaluated at [$]t_0[$], they are constants and we can write the above integrals as
[$]X(t)=X(t_0)+\mu (X(t_0)) \int_{t_0}^t d\tau[$]
[$]+\frac{\partial }{\partial X}[\mu ( X(t_0))]\mu (X(t_0)) \int_{t_0}^t \int_{t_0}^\tau \text{ds} d\tau[$]
[$]+ \frac{\partial }{\partial X}[\mu (X(t_0))]\sigma (X(t_0)) \int_{t_0}^t \int_{t_0}^\tau \text{dz(s)} d\tau[$]
[$]+0.5 \frac{\partial ^2[\mu ( X(t_0))]}{\partial X^2} \sigma ( X(t_0))^2 \int_{t_0}^t \int_{t_0}^\tau \text{ds} d\tau[$]
[$]+\sigma(X(t_0)) \int _{t_0}^t dz(\tau )[$]
[$]+\frac{\partial }{\partial X}[\sigma ( X(t_0))]\mu (X(t_0)) \int _{t_0}^t \int_{t_0}^\tau \text{ds} dz(\tau )[$]
[$]+\frac{\partial }{\partial X}[\sigma ( X(t_0))]\sigma (X(t_0))\int _{t_0}^t \int_{t_0}^\tau dz(s ) dz(\tau )[$]
[$]+0.5 \frac{\partial ^2[\sigma ( X(t_0))]}{\partial X^2} \sigma ( X(t_0))^2\int_{t_0}^t \int_{t_0}^\tau \text{ds} dz(\tau )[$] Equation(10)

The integrals of the kind [$]\int_{t_0}^t \int_{t_0}^\tau \text{ds} d\tau[$], [$]\int_{t_0}^t \int_{t_0}^\tau \text{dz(s)} d\tau[$] and [$]\int _{t_0}^t \int_{t_0}^\tau dz(s ) dz(\tau )[$] and other higher order integrals can be very easily analytically solved for monte carlo. These integrals also commute. I will post a monte carlo simulation code for CEV and lognormal process based on the above logic in a few days.

If you like my above posts, I am looking for consulting and contract work and you could contact me for that at my email anan2999(at)yahoo(dot)com

In the above post I mentioned that I can solve iterated dz and dt integrals which was not true(as I know now). I very embarassingly thought that these integrals commute but they really do not.
On Sun Mar 04, 2018 11:43 am, I made post#645 below when I posted the paper on monte carlo of SDEs on SSRN. here is the link to my paper: https://papers.ssrn.com/sol3/papers.cfm?abstract_id=3133959 I want to tell friends that I was on very high medication and I was not able to work very clearly. Still the equation in the paper explain the idea behind higher order monte carlo very well but the paper still has the mistake that I assumed that dz and dt integrals commute. This is an error in the paper that I really did not know at that time. Apart from this error, the ideas in the paper are fine.

Amin:
Here I uploaded my paper https://papers.ssrn.com/sol3/papers.cfm?abstract_id=3133959
The title is: Method of Iterated Integrals for Solution of Stochastic Integrals with Applications to Monte Carlo Simulation of Stochastic Differential Equations.
Abstract:
In this article we suggest a new method for solutions of stochastic integrals where the dynamics of the variables in integrand are given by some stochastic differential equation. We also propose numerical simulation of stochastic differential equations which is based on iterated integrals method employing Ito change of variable expansion of the drift and volatility function of the SDE. We repeatedly apply the Ito change of variable formula or Ito product rule to integrand and find its higher order derivatives and later we find the contribution from these higher order derivatives to the integrand using higher order iterated integrals. First order derivatives of the integrand, for example, would employ second order iterated integrals to find their contribution to the solution. We evaluate these iterated integrals terms at the initial time of the simulation along with the solution of the stochastic integrals which is found in terms of Hermite polynomials and variance of the integrals. We apply the method of iterated integrals to simulation of various stochastic differential equations. We find that the accuracy of simulation sharply increases using the method of iterated integral as compared to naïve and simple monte carlo simulation methods.

It was only later when I realized that I had made an error and dz and dt type integrals do not commute after I had run high order simulations and my results were not good. Here is a post#689 I made on Sun May 06, 2018 5:48 pm acknowledging the error in my work.

Amin:

I will take a few more days to complete the paper but here are a few formulas for friends. Here I give the general algorithm for the type of integrals where only dz and dt integrals follow each other in any order. Like many other possibilities, one four order repeated integral with two dz uncertainties could be[$]\int _0^t \text{$\sigma $dz}(s)\int _0^s\text{du}\int _0^u\text{dv}\int _0^v\sigma \text{dz}(w)[$]The value of the above kind of integrals, as long as there is no explicit t, does not depend upon the sequence order(whether dz integral comes first or dt integral comes first does not change the value of the total integrals) so variance can be easily calculated by Ito Isometry. The order of hermite polynomial depends upon the number of normal uncertainties so is equal to number of dz integrals. Here is the simple formula to calculate the hermite polynomial representation of the integrals which equals[$] H_n(N)*\frac{\sqrt{\text{Variance}}}{\sqrt{n!}}[$]here n, which is the order of hermite polynomials, equals the number of normal uncertainties. I will give a few examples[$]\int _0^t \text{$\sigma $dz}(s)\int _0^s\text{du}=H_1(N)\frac{\text{$\sigma $t}^{1.5}}{\sqrt{1!}\sqrt{3}}[$][$]\int _0^t \text{$\sigma $dz}(s)\int _0^s\text{$\sigma $dz}(u)=H_2(N)\frac{\sigma ^2t}{\sqrt{2!}\sqrt{2}}[$][$]\int _0^t \text{$\sigma $dz}(s)\int _0^s\text{du}\int _0^u\text{$\sigma $dz}(v)=\int _0^t \text{$\sigma $dz}(s)\int _0^s\text{$\sigma $dz}(u)\int _0^u\text{dv}=\int _0^t \text{ds}\int _0^s\text{$\sigma $dz}(u)\int _0^u\text{$\sigma $dz}(v)=H_2(N)\frac{\sigma ^2t^2}{\sqrt{2!}\sqrt{12}}[$][$]\int _0^t \text{$\sigma $dz}(s)\int _0^s\text{du}\int _0^u\text{dv}=\int _0^t \text{ds}\int _0^s\text{$\sigma $dz}(u)\int _0^u\text{dv}=\int _0^t \text{ds}\int _0^s\text{du}\int _0^u\text{$\sigma $dz}(v)=H_1(N)\frac{\sigma ^2t^{2.5}}{\sqrt{1!}\sqrt{20}}[$][$]\int _0^t \text{$\sigma $dz}(s)\int _0^s\text{$\sigma $dz}(u)\int _0^u\text{$\sigma $dz}(v)=H_3(N)\frac{\sigma ^3t^{1.5}}{\sqrt{3!}\sqrt{6}}[$]Fourth order and higher integrals can be evaluated to perfect precision using this method so I am not giving examples for that.
Ok friends, I believe that this is an error and all integrals do not commute. I will soon be posting a new program to calculate these integrals in a nice algorithmic fashion. But just want to warn friends that this is a mistake and will put the results slightly off.
Calculation of coefficients of stochastic integrals in an algorithmic fashion is perfectly fine but variance and calculations of related stochastic integrals is slightly off in many cases.

I thought about the ideas again and I was able to think of a parallel between [$]\int Zdt = tZ - \int t dz[$] and [$]\int H_n dt = t H_n - \int t nH_{n-1} dz[$] and that is how I solved the stochastic integrals. Here is a post#690 I made after four days on Thu May 10, 2018 10:38 pm

I was able to find these very nice recursions for integrals of products of time powers and hermite polynomials of various orders. Using this I get very perfect match when I expand using Ito-Taylor the powers of lognormal and time integrals of squared lognormal etc. I will soon be posting new matlab and mathematica programs with the corrections made for integrals. And calculations are no more complex than what we did before. But for friends who would be interested, these are the recursions for various stochastic integrals of hermite polynomials and their time products. With these recursions, we can very easily calculate all the stochastic integrals that arise in Ito-Taylor expansions whatever the order of integrals.

\int_0^t s^m H_n (z(s)) ds = t^(m+1)H_n (z(t))/( m+1)(1-sqrt(n)/sqrt(2m+n+2))

\int _0^t s^m H_n(z(s)) dz(s)=t^m H_(n+1)(z(t))/sqrt(n+1)/sqrt(2m+n+1)

Here H_n(z(t)) is hermite polynomial of degree n and H_(n+1)(z(t)) is hermite polynomial of degree n+1.

Here is another past where I latexed the integrals. This post # 693 was written on Fri May 11, 2018 6:08 pm,

Here I have latexed the update formulas for stochastic integrals

[$]\int_0^t s^m H_{n}(z(s)) ds = \frac {t^{m+1} H_{n}(z(t))}{(m+1)} (1 - \frac {\sqrt{n}}{\sqrt{(2m+n+2)}}) [$]

[$]\int_0^t s^m H_{n}(z(s)) dz(s)= \frac {t^m H_{n+1}(z(t))}{ \sqrt{(n+1)} \sqrt{(2m+n+1)}} [$]

Here is a post # 696, I made a few days later on Sun May 13, 2018 8:08 pm,

Amin:
I have two questions so as to have clarity about my contribution to literature.

a) Has somebody systematically solved these stochastic integrals that I wrote in post 693 before? I was reading some recent research papers and I noticed that authors did first few integrals with hand with a lot of effort. Would like to know if there is any reference about these integrals. Can somebody help with the literature review here? Of course, would be good to claim ownership if I have done them first so as to obviate the possibility of others claiming the same thing. But surely some expert literature review would be helpful. I believe any stochastic integral of Brownian motion can be solved with the equations in post 693.

b) Also I mostly see applications of Ito-Taylor expansions to monte carlo simulations of SDEs in almost all of the research papers. I have presented a framework in this thread to analytically calculate the density of SDEs and their stochastic integrals and relate these derived densities to density of standard normal random variable. Have some other people done that before?

I also see some contribution in writing a simple algorithm to calculate the Ito-Taylor coefficients in an efficient manner without resorting to symbolic computing but that is probably more basic. But I believe there is enough original material now to put together a very good research paper after throwing in some interesting , useful and practical examples that are based on analytic solution of densities of stochastic integrals of SDEs and their applications to mathematical finance.

And I made a lot of errors on the way. Thanks to all friends for pardoning and overlooking the mistakes.

At this time I had completed all the basic theoretical work for higher order monte carlo of SDEs and I was posting simple examples of higher order monte carlo.
Here in post# 697 written on Tue May 15, 2018 6:29 pm , I explained how to use the above recursion formulas for solution of iterated stochastic integrals.

Amin:
Here I have latexed the update formulas for stochastic integrals

[$]\int_0^t s^m H_{n}(z(s)) ds = \frac {t^{m+1} H_{n}(z(t))}{(m+1)} (1 - \frac {\sqrt{n}}{\sqrt{(2m+n+2)}}) [$]

[$]\int_0^t s^m H_{n}(z(s)) dz(s)= \frac {t^m H_{n+1}(z(t))}{ \sqrt{(n+1)} \sqrt{(2m+n+1)}} [$]
As I suggested, we can calculate most stochastic integrals using this update. Let us suppose, we want to calculate
[$]\int_0^t z(s)^4 ds[$] Equation(1)
Here is the way to go. We have to do it in terms of hermite polynomials.
fourth hermite polynomial representation of Brownian motion goes like
[$]H_4(z(t)) = z(t)^4 -6 t z(t)^2 +3 t^2 [$] Eq(2)
and second hermite polynomial is
[$]H_2(z(t)) = z(t)^2 - t [$] Eq(3)
and we know from the quoted post
[$]\int_0^t H_4(z(s)) ds = t H_4(z(t)) (1- \frac{2}{\sqrt{6}} ) [$] Eq(4)
so from Eq(2) and Eq(4)
[$]\int_0^t z(s)^4 ds = t H_4(z(t)) (1 - \frac{2}{\sqrt{6}} ) + 6 \int_0^t s z(s)^2 ds - 3 \int_0^t s^2 ds [$] Eq(5)
In order to solve the second term, we take the second hermite polynomial and use the formula in quotes to calculate that
[$]\int_0^t s H_2(z(s)) ds = \frac{t^2 H_2(z(t))}{2} (1 - \frac{\sqrt{2}}{\sqrt{6}} ) [$] Eq(6)
so
[$]\int_0^t s z(s)^2 ds = \frac{t^2 H_2(z(t))}{2} ( 1 - \frac{\sqrt{2}}{\sqrt{6}} ) + \int_0^t s^2 ds [$] Eq(7)
and we substitute Eq(7) in Eq(5) to get
[$]\int_0^t z(s)^4 ds = t H_4(z(t)) ( 1 - \frac{2}{\sqrt{6}} ) + 3 t^2 H_2(z(t)) ( 1 - \frac{\sqrt{2}}{\sqrt{6}} ) + 3 \int_0^t s^2 ds [$]
so
[$]\int_0^t z(s)^4 ds = t H_4(z(t)) (1- \frac{2}{\sqrt{6}} ) + 3 t^2 H_2(z(t)) ( 1 - \frac{\sqrt{2}}{\sqrt{6}} ) + t^3 [$]
and we could take [$]t^3[$] common and have a representation in terms of standard normal. And then you can easily simulate the density of [$]\int_0^t z(s)^4 ds[$] or analytically calculate and plot its density.

Here is another post#736 written on Mon Jul 02, 2018 10:56 am in which I have given the functions to calculate Ito-Taylor coefficients for monte carlo. This is exactly the same function that I have been distributing as a dependency of monte carlo simulations for more than year and a half now. In this program there are two main functions. One is ItoTaylorCoeffsNew which calculates the coefficients for monte carlo simulation for each different stochastic differential equation with two drift terms. There is another sub-function ComputeIntegralCoeffs that calculates the coefficients associated with iterated stochastic integrals of hermite polynomials using the same algorithm that I gave in post 689 and 693 as the coefficients that are evaluated at starting time are assumed as constant multipliers for these integrals. I have also tried to explain the algorithm behind above two functions in this post 736. Here is the post
Edit:I resume this post into the next post since it has exceeded maximum characters allowable.

Amin · March 16th, 2020, 2:54 pm

Here I resume the previous post.
Here is another post#736 written on Mon Jul 02, 2018 10:56 am in which I have given the functions to calculate Ito-Taylor coefficients for monte carlo. This is exactly the same function that I have been distributing as a dependency of monte carlo simulations for more than year and a half now. In this program there are two main functions. One is ItoTaylorCoeffsNew which calculates the coefficients for monte carlo simulation for each different stochastic differential equation with two drift terms. There is another sub-function ComputeIntegralCoeffs that calculates the coefficients associated with iterated stochastic integrals of hermite polynomials using the same algorithm that I gave in post 689 and 693 as the coefficients that are evaluated at starting time are assumed as constant multipliers for these integrals. I have also tried to explain the algorithm behind above two functions in this post 736. Here is the post

The program has the right algorithm for linear-like SDEs but my algorithm using  sqrt(1/T) *N for each step of normal density evolution probably does not seem to hold for very non-linear SDEslike stoch vol SDEs.
But I am giving the simulation algorithm in the hope that it would be helpful to provide a simple framework for people to experiment on their own and try their ideas where Ito-Taylor coefficients and simulation program has been calculated and normal variance structure or hermite polynomial type might have to be altered. My first thought would be that trying Brownian motion based hermite polynomials (orthogonal hermite polynomials that take Brownian motion variance) might really fix the problem with very non-linear SDEs and I hope that this simulation framework would be helpful for people to try brownian motion based hermite polynomials and other ideas they have. And it would be easier for many people to slightly change the existing program and try the ideas they have.
I have not been able to work for past few weeks for many reasons and would love to try many of these ideas on my own. But, of course, my personal inability to work must not stop other friends from working on their research ideas. And I would request friends to give me credit for the original basic work I have already done and would absolutely welcome all good research from other friends.

Here is the basic program explanation.
In the expansion of X(t)^alpha
With SDE
dX(t)=mu1 * X(t)^beta1 dt + mu2* X(t)^beta2 dt + sigma * X(t)^gamma dz(t)
The first order expansion of X^alpha with the above SDE is
X(t+1)^alpha= X(t)^alpha +alpha* X(t)^(alpha-1) * mu1 * X(t)^beta1 *dt
+alpha*X(t)^(alpha-1) * mu2 * X(t)^beta2*dt
+ .5* alpha *(alpha-1)* X(t)^(alpha-2) * sigma^2 * X(t)^(2*gamma) *dt
+ alpha * X(t)^(alpha-1) * sigma * X(t)^gamma *dz(t)
In my program, for the above four term expansion, I use a four dimensional array notation that shows the order of differentiation on each of these four types of terms. So if l1 represents drift one, l2 represents drift two and l3 represents quadratic variation term and l4 represents volatility term, we can use the notation Y(l1,l2,l3,l4) for the coefficients of expansion of the above term. So in the above first order expansion
Y(1,0,0,0)=alpha * mu1
Y(0,1,0,0)=alpha *mu2
Y(0,0,1,0)=.5* alpha *(alpha-1)* sigma^2
Y(0,0,0,1)= alpha * sigma

For higher orders of expansion, we will have to carefully compute the coefficients separately and then lump them together in Y according to array indices. Once we have done that, we can write one term of the general expansion as
Y(l1,l2,l3,l4) * x^(alpha+l1 * beta1 +l2* beta2 +l3* 2* gamma +l4 * gamma
- l1 -l2 - 2*l3 -l4) * HermiteP(l4) * dt^(l1 + l2 +l3 + .5*l4);

In my program, since matlab does not take indices starting from zero, I have slightly adjusted the above notation. In the density simulation program, we pre-compute everything but we have to calculate the part of above expression where power of x is evaluated.
If we have four terms in Ito-hermite SDE expansion at first order, In every next level, each of the four terms would take four descendent terms and there will be sixteen terms on second level, and on third level there will be 64 descendent terms and 256 descendent terms on fourth level. However we can reduce the number of terms in Ito-hermite expansion to number of terms in polynomial expansion. For example Ito-hermite expansion of SDE with four terms on first order can be reduced to the number of terms in expansion of a polynomial with four terms on each level. To do this, we will have to pre-calculate the Coefficients carefully and the lump them together as in a multinomial like expansion.
Again the above array Y(l1,l2,l3,l4) is relatively sparse and does not take all possible indices. Just like in polynomial expansion, on a certain expansion order, it takes only indices so that sum of indices goes to expansion order. For example on second order, only those indices could be found such that l1+l2+l3+l4=2. And similarly for third ito-hermite expansion level only those indices would be populated such that l1+l2+l3+l4=3.
In order to do that I have done simple loops in a fashion shown below

for k = 0 : (Order4)
for m = 0:k
        l4 = k - m;
for n = 0 : m
            l3 = m - n;
for j = 0:n
                l2 = n - j;
                l1 = j;
First loop variable is for expansion order. And then rest of the loop variables take values such that their sum equals the expansion order. When we would add a fifth term in the SDE expansion for example, we will have to add another loop variable with the same logic that sum of loop variables equal to expansion order. In my matlab program, I have given offset to l4, l2,l3, l1 since array indices cannot start at zero.

In the coefficient calculation program which calculates Y(l1,l2,l3,l4), I have used four levels of looping each for relevant expansion order. The first loop takes four values and second loop takes 16 values and third loop takes 64 values and so on. And then each coefficient term can be individually calculated while carefully accounting for path dependence.

So for example in a nested loop structure

m1= 1:mDim
m2=1:mDim
m3=1:mDim

l(m1)=l(m1)+1;
l(m2)=l(m2)+1;
l(m3)=l(m3)+1;
Y(l(1)+1,l(2),l(3),l(4))=Y(l(1)+1,l(2),l(3),l(4))+Coeff1st*IntegralCoeff(1,1,1,2);

in the above looping loop takes values from one to four with one indicating the first drift term, two indicating the second drift term and three indicating quadratic variation term and four indicating the volatility term. And with this looping structure we can
So in the above looping m1=1 would mean that all terms are descendent of first drift term and m2=4 would mean that all terms are descendent of first drift term on first expansion order and descendent of volatility term on the second order and so we can keep track of path dependence perfectly.

And on each level, we individually calculate the descendent terms. While keeping track of path dependence and calculating the coefficients with careful path dependence consideration, we update the appropriate element in our polynomial like expansion coefficientarray .Here l(1) denotes l1 but written as l(1) so it can be conveniently updated with the loop variable when the loop variable takes value one indicating first drift term . And l(2) could be conveniently updated when the loop variable takes value equal to two indicating second drift term and so on.

Here is the part of code snippet for that

form1=1:mDim
l(1)=1;
l(2)=1;
l(3)=1;
l(4)=1;
l(m1)=l(m1)+1;
CoeffDX1 = alpha + (l(1)-1) *beta1 + (l(2)-1) *beta2 + (l(3)-1) *2*gamma + (l(4)-1)*gamma - (l(1)-1) - (l(2)-1) - 2*(l(3)-1) - (l(4)-1);
    CoeffDX2 = CoeffDX1 - 1;
    ArrIndex0=m1;
ArrIndex=(m1-1)*mDim;
    Coeff1st=Y1(ArrIndex0)*CoeffDX1;
    Coeff2nd=Y1(ArrIndex0)*.5*CoeffDX1*CoeffDX2;
Y2(1+ArrIndex)=Coeff1st;
    Y(l(1)+1,l(2),l(3),l(4))=Y(l(1)+1,l(2),l(3),l(4))+Coeff1st*IntegralCoeff(1,1,2,n1(m1));
Y2(2+ArrIndex)=Coeff1st;
    Y(l(1),l(2)+1,l(3),l(4))=Y(l(1),l(2)+1,l(3),l(4))+Coeff1st*IntegralCoeff(1,1,2,n1(m1));
Y2(3+ArrIndex)=Coeff2nd;
    Y(l(1),l(2),l(3)+1,l(4))=Y(l(1),l(2),l(3)+1,l(4))+Coeff2nd*IntegralCoeff(1,1,2,n1(m1));
Y2(4+ArrIndex)=Coeff1st;
    Y(l(1),l(2),l(3),l(4)+1)=Y(l(1),l(2),l(3),l(4)+1)+Coeff1st*IntegralCoeff(1,1,3,n1(m1));

The first four lines update the array indices according to the parent term. And then CoeffDX1 and CoeffDX2 are calculated according to algebraic exponents on parent terms.
ArrIndex0=m1;calculates the array index of the parent term
And
ArrIndex=(m1-1)*mDim;calculates the array index of the descendent terms

And coefficient of the drift and volatility descendent terms is calculated by multiplying the coefficient of the parent term by
Coeff1st=Y1(ArrIndex0)*CoeffDX1;

And coefficient of the quadratic variation descendent terms is calculated by multiplying the coefficient of the parent term by

Coeff2nd=Y1(ArrIndex0)*.5*CoeffDX1*CoeffDX2;

And then each of the four descendent terms are updated with Coeff1st if they are drift or volatility descendent terms or Coeff2nd if they are quadratic variation descendent terms.
Here Y1 indicates the temporary coefficient array with parent terms on first level. And Y2 denotes temporary coefficient array with parent terms on second level and Y3 indicates temporary coefficient array with parent terms on third level.

Here is the main simulation Code after the coefficients have been calculated

function [] = ItoTaylorDensitySimulation()


% In the expansion of X^alpha
% With SDE
% dX(t)=mu1 * X(t)^beta1 dt + mu2* X(t)^beta2 dt + sigma * X(t)^gamma dz(t)
% The first order expansion of X^alpha with the above SDE is 
% X(t+1)^alpha= X(t)^alpha +alpha* X(t)^(alpha-1) * mu1 * X(t)^beta1 *dt 
% +alpha* X(t)^(alpha-1) * mu2 * X(t)^beta2 *dt 
% + .5* alpha *(alpha-1)* X(t)^(alpha-2) * sigma^2 * X(t)^(2*gamma) *dt
% + alpha * X(t)^(alpha-1) * sigma * X(t)^gamma *dz(t)

% In my program, for the above four term expansion, I use a four dimensional 
%array notation that shows the order of differentiation on each of these 
%four types of terms. So if l1 represents drift one, l2 represents drift 
%two and l3 represents quadratic variation term and l4 represents 
%volatility term, we can use the notation Y(l1,l2,l3,l4) for the 
%coefficients of expansion of the above term. So in the above first order expansion 

% Y(1,0,0,0)= alpha * mu1 
% Y(0,1,0,0)=alpha *mu2
% Y(0,0,1,0)= .5* alpha *(alpha-1)* sigma^2 
% Y(0,0,0,1)= alpha * sigma
% 
% For higher orders of expansion, we will have to carefully compute the 
%coefficients separately based on path dependence and then lump them 
%together in Y according to array indices. Once we have done that, 
%we can write one term of the general expansion as 

% Y(l1,l2,l3,l4) * x^(alpha+l1 * beta1 +l2* beta2 +l3* 2* gamma +l4 * gamma
%  - l1 -l2 - 2*l3 -l4) * HermiteP(l4) * dt^(l1 + l2 +l3 + .5*l4);
% 
% In my program, since matlab does not take indices starting from zero, 
%I have slightly adjusted the above notation.  
%In the density simulation program, we pre-compute all coefficients and values
%needed in simulation but we have to calculate the part of above 
%expression where power of x is evaluated.
% If we have four terms in Ito-hermite SDE expansion at first order, 
%In every next level, each of the four terms would take four descendent 
%terms and there will be sixteen terms on second level, and on third level 
%there will be 64 descendent terms and 256 descendent terms on fourth level. 
%However we can reduce the number of terms in Ito-hermite expansion to 
%number of terms in polynomial expansion. For example Ito-hermite expansion
%of SDE with four terms on first order can be reduced to the number of 
%terms in expansion of a polynomial with four terms on each level. 
%To do this, we will have to pre-calculate the Coefficients carefully and 
%the lump them together as in a polynomial like expansion. 
% Again the above array Y(l1,l2,l3,l4) is relatively sparse and 
%does not take all possible indices. Just like in polynomial expansion,
%on a certain expansion order, it takes only indices so that sum of indices
%goes to expansion order. For example on second order, only those indices 
%could be found such that l1+l2+l3+l4=2. And similarly for third ito-hermite 
%expansion level only those indices would be populated such that l1+l2+l3+l4=3. 
% In order to do that I have done simple loops in a fashion shown below
% 
% for k = 0 : (Order4)
%     for m = 0:k
%         l4 = k - m;
%         for n = 0 : m
%             l3 = m - n;
%             for j = 0:n
%                 l2 = n - j;
%                 l1 = j;
% First loop variable is for expansion order. And then rest of the loop 
%variables take values such that their sum equals the expansion order. 
%When we would add a fifth term in the SDE expansion for example, 
%we will have to add another loop variable with the same logic that 
%sum of loop variables equal to expansion order. In my matlab program, 
%I have given offset to l4, l2,l3, l1 since array indices cannot start at zero. 


Order4=4;  %expansion order
Nn=161;    % No of normal density subdivisions
dNn=.05;   % Normal density width. would change with number of subdivisions
dt=.250;   % Simulation time interval.
Tt=2;      % simulation level. Terminal time= Tt*dt;  
x0=.250;   % starting value of SDE


alpha=1;   % x^alpha is being expanded
beta1=0;   % the first drift term power.
beta2=1;   % Second drift term power.
gamma=1;   % volatility power.
mu1=.25;   % first drift coefficient.
mu2=-1;    % Second drift coefficient.
sigma0=.750;%Volatility value

for nn=1:Nn
    x(nn)=x0;
end

for nn=1:Nn
    Z(nn)=((nn-1)*dNn-4.0)*1/sqrt(Tt);
    
end

ZProb(1)=normcdf(.5*Z(1)+.5*Z(2),0,1/sqrt(Tt));
ZProb(Nn)=1-normcdf(.5*Z(Nn)+.5*Z(Nn-1),0,1/sqrt(Tt));
for nn=2:Nn-1
    ZProb(nn)=normcdf(.5*Z(nn)+.5*Z(nn+1),0,1/sqrt(Tt))-normcdf(.5*Z(nn)+.5*Z(nn-1),0,1/sqrt(Tt));
end

%ZProb contains probaility mass in each cell.

% Pre-compute the hermite polynomilas associated with each subdivision of
% normal.
for nn=1:Nn
        HermiteP(1,nn)=1;
        HermiteP(2,nn)=Z(nn);
        HermiteP(3,nn)=Z(nn)^2-1;
        HermiteP(4,nn)=Z(nn)^3-3*Z(nn);
        HermiteP(5,nn)=Z(nn)^4-6*Z(nn)^2+3;
end

%These sigma11, mu11, mu22, sigma22 can be lumped with Y(l1,l2,l3,l4) but I
%kept them separate for a general program where these values could change
%on every step while exponents on x in each term would remain same in
%pre-calcualted Y(l1,l2,l3,l4)

sigma11(1:Order4+1)=0;
mu11(1:Order4+1)=0;
mu22(1:Order4+1)=0;
sigma22(1:Order4+1)=0;

% index 1 correponds to zero level since matlab indexing starts at one. 
sigma11(1)=1;
mu11(1)=1;
mu22(1)=1;
sigma22(1)=1;

for k=1:(Order4+1)
    if sigma0~=0
        sigma11(k)=sigma0^(k-1);
    end
    if mu1 ~= 0
        mu11(k)=mu1^(k-1);
    end
    if mu2 ~= 0
        mu22(k)=mu2^(k-1);
    end
    if sigma0~=0
        sigma22(k)=sigma0^(2*(k-1));
    end
end

Ft(1:Tt+1,1:(Order4+1),1:(Order4+1),1:(Order4+1),1:(Order4+1))=0;
Fp(1:(Order4+1),1:(Order4+1),1:(Order4+1),1:(Order4+1))=0;

%Pre-compute the time and power exponent values in small multi-dimensional arrays

for k = 0 : (Order4)
    for m = 0:k
        l4 = k - m + 1;
        for n = 0 : m
            l3 = m - n + 1;
            for j = 0:n
                l2 = n - j + 1;
                l1 = j + 1;
                Ft(l1,l2,l3,l4) = dt^((l1-1) + (l2-1) + (l3-1) + .5* (l4-1));
                Fp(l1,l2,l3,l4) = (alpha + (l1-1) * beta1 + (l2-1) * beta2 + (l3-1) * 2* gamma + (l4-1) * gamma ...
                    - (l1-1) - (l2-1) - 2* (l3-1) - (l4-1));
            end
        end
    end
end

%Below call the program that calculates the Ito-Taylor coeffs. 

Y = ItoTaylorCoeffsNew(alpha,beta1,beta2,gamma);
%YdtCoeff= ItoTaylorCoeffsNew(alphadt,beta1,beta2,gamma);
%YdzCoeff= ItoTaylorCoeffsNew(alphadz,beta1,beta2,gamma);
Y(1,1,1,1)=0;

for tt=1:Tt
    for nn=1:Nn
        x1(nn)=x(nn);
        for k = 0 : Order4
            for m = 0:k
                l4 = k - m + 1;
                for n = 0 : m
                    l3 = m - n + 1;
                    for j = 0:n
                        l2 = n - j + 1;
                        l1 = j + 1;
                        x1(nn)=x1(nn) + Y(l1,l2,l3,l4) * ...
                            mu11(l1)*mu22(l2)*sigma22(l3)*sigma11(l4)* ...
                            x(nn)^Fp(l1,l2,l3,l4) * HermiteP(l4,nn) * Ft(l1,l2,l3,l4);
                    end
                end
            end
        end
    x(nn)=x1(nn);
    x(nn)
    end
end



for nn=1:Nn
    B(nn)=((nn-1)*dNn-4.0);
end

for nn=2:Nn-1
    Dfx(nn) = (x(nn + 1) - x(nn - 1))/(B(nn + 1) - B(nn - 1));
end
fx(1:Nn)=0;
for nn = 2:Nn-1
    fx(nn) = normpdf(B(nn),0, 1)/abs(Dfx(nn));
end


%x

%Ft(1,1,1,2)
%Calculate the density with monte carlo below.

dt0=dt/8;
paths=100000;
X(1:paths)=x0;
Random1(1:paths)=0;
for tt=1:Tt*8
    Random1=randn(size(Random1));
    X(1:paths)=X(1:paths)+ mu1* X(1:paths).^beta1 *dt0 + ...
        mu2 * X(1:paths).^beta2*dt0 + ...
        sigma0 * X(1:paths).^gamma .* Random1(1:paths) *sqrt(dt0) + ...
        mu1^2 * beta1* X(1:paths).^(2*beta1-1) *  dt0^2/2.0 + ...
        mu2^2 * beta2 * X(1:paths).^(2*beta2-1) * dt0^2/2 + ...
        mu1*mu2* beta1* X(1:paths).^(beta1+beta2-1) *dt0^2/2.0 + ...
        mu1*mu2* beta2* X(1:paths).^(beta1+beta2-1) *dt0^2/2.0 + ...
        sigma0^2 * gamma * X(1:paths).^(2*gamma-1) .* dt0/2.0 .* (Random1(1:paths).^2-1) + ... 
        mu1 * sigma0 *gamma* X(1:paths).^(beta1+gamma-1) .* sqrt(1/3) .* dt0^1.5 .* Random1(1:paths) + ...
        mu2 * sigma0 *gamma* X(1:paths).^(beta2+gamma-1) .* sqrt(1/3) .* dt0^1.5 .* Random1(1:paths) + ...
        sigma0 * mu1 *beta1* X(1:paths).^(beta1+gamma-1) .* (1-sqrt(1/3)) .* dt0^1.5 .* Random1(1:paths) + ...
        sigma0 * mu2.*beta2* X(1:paths).^(beta2+gamma-1) .* (1-sqrt(1/3)) .* dt0^1.5 .* Random1(1:paths);
end

sum(X(:))/paths   %monte carlo average
sum(x(:).*ZProb(:))  % Ito-Taylor density average(good enough but slightly approximate)

BinSize=.00125;
MaxCutOff=20;
[XDensity,IndexOutX,IndexMaxX] = MakeDensityFromSimulation_Infiniti(X,paths,BinSize,MaxCutOff);
plot(x(2:Nn-1),fx(2:Nn-1),'r',IndexOutX(1:IndexMaxX),XDensity(1:IndexMaxX),'g');

end

Here is the Coefficient calculation program

function [Y] = ItoTaylorCoeffsNew(alpha,beta1,beta2,gamma)


%In the coefficient calculation program which calculates Y(l1,l2,l3,l4), 
%I have used four levels of looping each for relevant expansion order. 
%The first loop takes four values and second loop takes 16 values and 
%third loop takes 64 values and so on. And then each coefficient 
%term can be individually calculated while carefully accounting 
%for path dependence. 
%So for example in a nested loop structure 
%m1= 1:mDim
% m2=1:mDim
% m3=1:mDim
%    l(m1)=l(m1)+1;
%    l(m2)=l(m2)+1;
%    l(m3)=l(m3)+1;

%in the above looping loop takes values from one to four with one 
%indicating the first drift term, two indicating the second drift term 
%and three indicating quadratic variation term and 
%four indicating the volatility term. And with this looping structure 
%we can So in the above looping m1=1 would mean that all terms are 
%descendent of first drift term and m2=4 would mean that all terms are 
%descendent of first drift term on first expansion order and descendent 
%of volatility term on the second order and so we can keep track of path 
%dependence perfectly. 
%And on each level, we individually calculate the descendent terms. While 
%keeping track of path dependence and calculating the coefficients with 
%careful path dependence consideration, we update the appropriate element 
%in our polynomial like expansion coefficient array 

%explaining the part of code
%m1= 1:mDim
% m2=1:mDim
% m3=1:mDim
%    l(m1)=l(m1)+1;
%    l(m2)=l(m2)+1;
%    l(m3)=l(m3)+1;
%Y(l(1)+1,l(2),l(3),l(4))=Y(l(1)+1,l(2),l(3),l(4))+Coeff1st*IntegralCoeff(1,1,1,2);

%Here l(1) denotes l1 but written as l(1) so it can be conveniently 
%updated with the loop variable when the loop variable takes value one 
%indicating first drift term . And l(2) could be conveniently updated when 
%the loop variable takes value equal to two indicating second 
%drift term and so on.
%Here is the part of code snippet for that

%for m1=1:mDim
%    l(1)=1;
%    l(2)=1;
%    l(3)=1;
%    l(4)=1;
%    l(m1)=l(m1)+1;
%CoeffDX1 = alpha + (l(1)-1) *beta1 + (l(2)-1) *beta2 + (l(3)-1) *2*gamma + (l(4)-1)*gamma - (l(1)-1) - (l(2)-1) - 2*(l(3)-1) - (l(4)-1);
%    CoeffDX2 = CoeffDX1 - 1;
%    ArrIndex0=m1;
%    ArrIndex=(m1-1)*mDim;
%    Coeff1st=Y1(ArrIndex0)*CoeffDX1;
%    Coeff2nd=Y1(ArrIndex0)*.5*CoeffDX1*CoeffDX2;
%    Y2(1+ArrIndex)=Coeff1st;
%    Y(l(1)+1,l(2),l(3),l(4))=Y(l(1)+1,l(2),l(3),l(4))+Coeff1st*IntegralCoeff(1,1,2,n1(m1));
%    Y2(2+ArrIndex)=Coeff1st;
%    Y(l(1),l(2)+1,l(3),l(4))=Y(l(1),l(2)+1,l(3),l(4))+Coeff1st*IntegralCoeff(1,1,2,n1(m1));
%    Y2(3+ArrIndex)=Coeff2nd;
%    Y(l(1),l(2),l(3)+1,l(4))=Y(l(1),l(2),l(3)+1,l(4))+Coeff2nd*IntegralCoeff(1,1,2,n1(m1));
%    Y2(4+ArrIndex)=Coeff1st;
%    Y(l(1),l(2),l(3),l(4)+1)=Y(l(1),l(2),l(3),l(4)+1)+Coeff1st*IntegralCoeff(1,1,3,n1(m1));

 
%The first four lines update the array indices according to the parent term. 
%And then CoeffDX1 and CoeffDX2 are calculated according to algebraic exponents on parent terms.    
%ArrIndex0=m1; calculates the array index of the parent term 
%And ArrIndex=(m1-1)*mDim; calculates the array index of the descendent terms
%And coefficient of the drift and volatility descendent terms is 
%calculated by multiplying the coefficient of the parent term by 
%Coeff1st=Y1(ArrIndex0)*CoeffDX1;
%And coefficient of the quadratic variation descendent terms is 
%calculated by multiplying the coefficient of the parent term by 
%Coeff2nd=Y1(ArrIndex0)*.5*CoeffDX1*CoeffDX2;
%And then each of the four descendent terms are updated with Coeff1st 
%if they are drift or volatility descendent terms or Coeff2nd if 
%they are quadratic variation descendent terms.
%Here Y1 indicates the temporary coefficient array with parent terms on 
%first level. And Y2 denotes temporary coefficient array with parent terms 
%on second level and Y3 indicates temporary coefficient array with parent terms on third level.

[IntegralCoeff,IntegralCoeffdt,IntegralCoeffdz] = ComputeIntegralCoeffs();

n1(1)=2;
n1(2)=2;
n1(3)=2;
n1(4)=3;
%n1(1), n1(2), n1(3) are drift and quadratic variation term variables 
%and take a value equal to 2 indicating a dt integral.
%n1(4) is volatility term variable and indicates a dz-integral by taking a
%value of 3. 

mDim=4; % four descendent terms in each expansion
Y(1:5,1:5,1:5,1:5)=0;
Y1(1:mDim)=0;
Y2(1:mDim*mDim)=0;
Y3(1:mDim*mDim*mDim)=0;


%First Ito-hermite expansion level starts here. No loops but four
%descendent terms.
l(1)=1;
l(2)=1;
l(3)=1;
l(4)=1;
CoeffDX1 = alpha;
CoeffDX2 = CoeffDX1 - 1;
Coeff1st=CoeffDX1;
Coeff2nd=.5*CoeffDX1*CoeffDX2;

Y1(1)=Coeff1st;
Y(l(1)+1,l(2),l(3),l(4))=Y(l(1)+1,l(2),l(3),l(4))+Coeff1st*IntegralCoeff(1,1,1,2);
Y1(2)=Coeff1st;
Y(l(1),l(2)+1,l(3),l(4))=Y(l(1),l(2)+1,l(3),l(4))+Coeff1st*IntegralCoeff(1,1,1,2);
Y1(3)=Coeff2nd;
Y(l(1),l(2),l(3)+1,l(4))=Y(l(1),l(2),l(3)+1,l(4))+Coeff2nd*IntegralCoeff(1,1,1,2);
Y1(4)=Coeff1st;
Y(l(1),l(2),l(3),l(4)+1)=Y(l(1),l(2),l(3),l(4)+1)+Coeff1st*IntegralCoeff(1,1,1,3);

%Second Ito-hermite expansion level starts. It has a loop over four parent
%terms and there are four descendent terms for each parent term.
%The coefficient terms are then lumped in a polynomial-like expansion
%array of coefficients.
for m1=1:mDim
    l(1)=1;
    l(2)=1;
    l(3)=1;
    l(4)=1;
    l(m1)=l(m1)+1;
    CoeffDX1 = alpha + (l(1)-1) *beta1 + (l(2)-1) *beta2 + (l(3)-1) *2*gamma + (l(4)-1)*gamma ...
                        - (l(1)-1) - (l(2)-1) - 2*(l(3)-1) - (l(4)-1);
    CoeffDX2 = CoeffDX1 - 1;
    ArrIndex0=m1;
    ArrIndex=(m1-1)*mDim;
    Coeff1st=Y1(ArrIndex0)*CoeffDX1;
    Coeff2nd=Y1(ArrIndex0)*.5*CoeffDX1*CoeffDX2;
    Y2(1+ArrIndex)=Coeff1st;
    Y(l(1)+1,l(2),l(3),l(4))=Y(l(1)+1,l(2),l(3),l(4))+Coeff1st*IntegralCoeff(1,1,2,n1(m1));
    Y2(2+ArrIndex)=Coeff1st;
    Y(l(1),l(2)+1,l(3),l(4))=Y(l(1),l(2)+1,l(3),l(4))+Coeff1st*IntegralCoeff(1,1,2,n1(m1));
    Y2(3+ArrIndex)=Coeff2nd;
    Y(l(1),l(2),l(3)+1,l(4))=Y(l(1),l(2),l(3)+1,l(4))+Coeff2nd*IntegralCoeff(1,1,2,n1(m1));
    Y2(4+ArrIndex)=Coeff1st;
    Y(l(1),l(2),l(3),l(4)+1)=Y(l(1),l(2),l(3),l(4)+1)+Coeff1st*IntegralCoeff(1,1,3,n1(m1));
    %Third Ito-hermite expansion level starts and it is a nested loop with
    %a total of sixteen parents and each parent takes four descendent
    %terms.
    for m2=1:mDim
        l(1)=1;
        l(2)=1;
        l(3)=1;
        l(4)=1;
        l(m1)=l(m1)+1;
        l(m2)=l(m2)+1;
        CoeffDX1 = alpha + (l(1)-1) *beta1 + (l(2)-1) *beta2 + (l(3)-1) *2*gamma + (l(4)-1)*gamma ...
                        - (l(1)-1) - (l(2)-1) - 2*(l(3)-1) - (l(4)-1);
        CoeffDX2=CoeffDX1-1;            
        ArrIndex0=(m1-1)*mDim+m2;
        ArrIndex=((m1-1)*mDim+(m2-1))*mDim;
        Coeff1st=Y2(ArrIndex0)*CoeffDX1;
        Coeff2nd=Y2(ArrIndex0)*.5*CoeffDX1*CoeffDX2;
        Y3(1+ArrIndex)=Coeff1st;
        Y(l(1)+1,l(2),l(3),l(4))=Y(l(1)+1,l(2),l(3),l(4))+Coeff1st*IntegralCoeff(1,2,n1(m2),n1(m1));
        Y3(2+ArrIndex)=Coeff1st;
        Y(l(1),l(2)+1,l(3),l(4))=Y(l(1),l(2)+1,l(3),l(4))+Coeff1st*IntegralCoeff(1,2,n1(m2),n1(m1));
        Y3(3+ArrIndex)=Coeff2nd;
        Y(l(1),l(2),l(3)+1,l(4))=Y(l(1),l(2),l(3)+1,l(4))+Coeff2nd*IntegralCoeff(1,2,n1(m2),n1(m1));
        Y3(4+ArrIndex)=Coeff1st;
        Y(l(1),l(2),l(3),l(4)+1)=Y(l(1),l(2),l(3),l(4)+1)+Coeff1st*IntegralCoeff(1,3,n1(m2),n1(m1));
        %fourht Ito-hermite expansion level starts and it is a triply-nested loop with
        %a total of sixteen parents and each parent takes four descendent
        %terms. We then lump the terms in a relatively sparse polynomial 
        %like expansion coefficient array that has smaller number of
        %non-zero terms.

        for m3=1:mDim
            l(1)=1;
            l(2)=1;
            l(3)=1;
            l(4)=1;
            l(m1)=l(m1)+1;
            l(m2)=l(m2)+1;
            l(m3)=l(m3)+1;
            CoeffDX1 = alpha + (l(1)-1) *beta1 + (l(2)-1) *beta2 + (l(3)-1) *2*gamma + (l(4)-1)*gamma ...
                        - (l(1)-1) - (l(2)-1) - 2*(l(3)-1) - (l(4)-1);
            CoeffDX2=CoeffDX1-1;
            ArrIndex0=((m1-1)*mDim+(m2-1))*mDim+m3;
            %ArrIndex=(((m1-1)*mDim+(m2-1))*mDim+(m3-1))*mDim;
            Coeff1st=Y3(ArrIndex0)*CoeffDX1;
            Coeff2nd=Y3(ArrIndex0)*.5*CoeffDX1*CoeffDX2;
            %Y4(1+ArrIndex)=Coeff1st;
            Y(l(1)+1,l(2),l(3),l(4))=Y(l(1)+1,l(2),l(3),l(4))+Coeff1st*IntegralCoeff(2,n1(m3),n1(m2),n1(m1));
            %Y4(2+ArrIndex)=Coeff1st;
            Y(l(1),l(2)+1,l(3),l(4))=Y(l(1),l(2)+1,l(3),l(4))+Coeff1st*IntegralCoeff(2,n1(m3),n1(m2),n1(m1));
            %Y4(3+ArrIndex)=Coeff2nd;
            Y(l(1),l(2),l(3)+1,l(4))=Y(l(1),l(2),l(3)+1,l(4))+Coeff2nd*IntegralCoeff(2,n1(m3),n1(m2),n1(m1));
            %Y4(4+ArrIndex)=Coeff1st;
            Y(l(1),l(2),l(3),l(4)+1)=Y(l(1),l(2),l(3),l(4)+1)+Coeff1st*IntegralCoeff(3,n1(m3),n1(m2),n1(m1));
         end
    end
end
end

Here is the stochastic integrals calculation program

function [IntegralCoeff0,IntegralCoeffdt,IntegralCoeffdz] = ComputeIntegralCoeffs()


IntegralCoeff(1:3,1:3,1:3,1:3)=0;
IntegralCoeffdt(1:3,1:3,1:3,1:3)=0;
IntegralCoeffdz(1:3,1:3,1:3,1:3)=0;

IntegralCoeff0(1:3,1:3,1:3,1:3)=0;

IntegralCoeff0(1,1,1,2)=1;
IntegralCoeff0(1,1,1,3)=1;

IntegralCoeff0(1,1,2,2)=1/2;
IntegralCoeff0(1,1,3,2)=1-1/sqrt(3);
IntegralCoeff0(1,1,2,3)=1/sqrt(3);
IntegralCoeff0(1,1,3,3)=1/2;

IntegralCoeff0(1,2,2,2)=1/6;
IntegralCoeff0(1,3,2,2)=(1-1/sqrt(3))*1/2*(1-1/sqrt(5));
IntegralCoeff0(1,2,3,2)=1/sqrt(3)/2*(1-1/sqrt(5));
IntegralCoeff0(1,3,3,2)=1/2*(1-sqrt(2)/2);
IntegralCoeff0(1,2,2,3)=1/2*1/sqrt(5);
IntegralCoeff0(1,3,2,3)=(1-1/sqrt(3))*1/sqrt(2)*1/2;
IntegralCoeff0(1,2,3,3)=1/sqrt(3)/sqrt(2)*1/sqrt(4);
IntegralCoeff0(1,3,3,3)=1/6;

IntegralCoeff0(2,2,2,2)=1/24;
IntegralCoeff0(2,2,3,2)=1/2*1/sqrt(5)*1/3*(1-1/sqrt(7));
IntegralCoeff0(2,3,2,2)=1/sqrt(3)*1/2*(1-1/sqrt(5))*1/3*(1-1/sqrt(7));
IntegralCoeff0(3,2,2,2)=(1-1/sqrt(3))*1/2*(1-1/sqrt(5))*1/3*(1-1/sqrt(7));
IntegralCoeff0(3,3,2,2)=1/2*(1-sqrt(2)/2)*1/2*(1-sqrt(2)/sqrt(6));
IntegralCoeff0(2,3,3,2)=1/sqrt(3)*1/sqrt(2)*1/2*1/2*(1-sqrt(2)/sqrt(6));
IntegralCoeff0(3,2,3,2)=(1-1/sqrt(3))*1/sqrt(2)*1/2*1/2*(1-sqrt(2)/sqrt(6));
IntegralCoeff0(3,3,3,2)=1/6*(1-sqrt(3)/sqrt(5));

IntegralCoeff0(2,2,2,3)=1/6*1/sqrt(7);
IntegralCoeff0(2,2,3,3)=1/2*1/sqrt(5)*1/sqrt(2)*1/sqrt(6);
IntegralCoeff0(2,3,2,3)=1/sqrt(3)*1/2*(1-1/sqrt(5))*1/sqrt(2)*1/sqrt(6);
IntegralCoeff0(3,2,2,3)=(1-1/sqrt(3))*1/2*(1-1/sqrt(5))*1/sqrt(2)*1/sqrt(6);
IntegralCoeff0(3,3,2,3)=1/2*(1-sqrt(2)/2)*1/sqrt(3)*1/sqrt(5);
IntegralCoeff0(2,3,3,3)=1/sqrt(3)*1/sqrt(2)*1/sqrt(4)*1/sqrt(3)*(1/sqrt(5));
IntegralCoeff0(3,2,3,3)=(1-1/sqrt(3))*1/sqrt(2)*1/2*1/sqrt(3)*1/sqrt(5);
IntegralCoeff0(3,3,3,3)=1/24;


%Can also be calculated with algorithm below.
%here IntegralCoeffdt indicates the coefficients of a dt-integral.
%This dt-integral means that you are calculating the Ito-hermite expansion
%of  Integral X(t)^alpha dt with X(t) dynamics given by the SDE 
%here IntegralCoeffdz indicates the coefficients of a dz-integral.
%This dz-integral means that you are calculating the Ito-hermite expansion
%of  Integral X(t)^alpha dz(t) with X(t) dynamics given by the SDE 

%IntegralCoeff is is associated with expansion of X(t)^alpha.
%IntegralCoeff below is not returned and IntegralCoeff0 manually calculated
%above is returned but both are the same.

l0(1:2)=1;

for m4=1:2
    l0(1)=1;
    l0(2)=1;
    
    %IntegralCoeff4(m4,1,1,1)=1;
    %IntegralCoeff4(m4,1,1,1)=1;
    %1 is added to m4 since index 1 stands for zero, 2 for one and three
    %for two.
    IntegralCoeff(1,1,1,m4+1)=1;
    l0(m4)=l0(m4)+1;
    IntegralCoeffdt(1,1,1,m4+1)=IntegralCoeff(1,1,1,m4+1)* ...
        1/(l0(1)-1+1)*(1-sqrt(l0(2)-1)/sqrt(2*(l0(1)-1)+(l0(2)-1)+2));
    IntegralCoeffdz(1,1,1,m4+1)= IntegralCoeff(1,1,1,m4+1)* ...
        1/sqrt(l0(2)-1+1)/sqrt(2*(l0(1)-1)+(l0(2)-1)+1);
    for m3=1:2
        l0(1)=1;
        l0(2)=1;
        l0(m4)=l0(m4)+1;
        
        if(m3==1)
            IntegralCoeff(1,1,m4+1,m3+1)=1/(l0(1)-1+1)*(1-sqrt(l0(2)-1)/sqrt(2*(l0(1)-1)+(l0(2)-1)+2));
        end
        if(m3==2)
            IntegralCoeff(1,1,m4+1,m3+1)= 1/sqrt(l0(2)-1+1)/sqrt(2*(l0(1)-1)+(l0(2)-1)+1);
        end
        l0(m3)=l0(m3)+1;
        %IntegralCoeff(1,1,m4+1,m3+1)=IntegralCoeff4(m4,m3,1,1);
        IntegralCoeffdt(1,1,m4+1,m3+1)=IntegralCoeff(1,1,m4+1,m3+1)* ...
            1/(l0(1)-1+1)*(1-sqrt(l0(2)-1)/sqrt(2*(l0(1)-1)+(l0(2)-1)+2));
        IntegralCoeffdz(1,1,m4+1,m3+1)= IntegralCoeff(1,1,m4+1,m3+1)* ...
            1/sqrt(l0(2)-1+1)/sqrt(2*(l0(1)-1)+(l0(2)-1)+1);
        for m2=1:2
            l0(1)=1;
            l0(2)=1;
            l0(m4)=l0(m4)+1;
            l0(m3)=l0(m3)+1;
            
            if(m2==1)
                IntegralCoeff(1,m4+1,m3+1,m2+1)=IntegralCoeff(1,1,m4+1,m3+1)*1/(l0(1)-1+1)*(1-sqrt(l0(2)-1)/sqrt(2*(l0(1)-1)+(l0(2)-1)+2));
            end
            if(m2==2)
                IntegralCoeff(1,m4+1,m3+1,m2+1)= IntegralCoeff(1,1,m4+1,m3+1)*1/sqrt(l0(2)-1+1)/sqrt(2*(l0(1)-1)+(l0(2)-1)+1);
            end
            l0(m2)=l0(m2)+1;
            %IntegralCoeff(1,m4+1,m3+1,m2+1)=IntegralCoeff4(m4,m3,m2,1);
            IntegralCoeffdt(1,m4+1,m3+1,m2+1)=IntegralCoeff(1,m4+1,m3+1,m2+1)* ...
                1/(l0(1)-1+1)*(1-sqrt(l0(2)-1)/sqrt(2*(l0(1)-1)+(l0(2)-1)+2));
            IntegralCoeffdz(1,m4+1,m3+1,m2+1)= IntegralCoeff(1,m4+1,m3+1,m2+1)* ...
                1/sqrt(l0(2)-1+1)/sqrt(2*(l0(1)-1)+(l0(2)-1)+1);

            for m1=1:2
                l0(1)=1;
                l0(2)=1;
                l0(m4)=l0(m4)+1;
                l0(m3)=l0(m3)+1;
                l0(m2)=l0(m2)+1;
                if(m1==1)
                    IntegralCoeff(m4+1,m3+1,m2+1,m1+1)=IntegralCoeff(1,m4+1,m3+1,m2+1)*1/(l0(1)-1+1)*(1-sqrt(l0(2)-1)/sqrt(2*(l0(1)-1)+(l0(2)-1)+2));
                end
                if(m1==2)
                    IntegralCoeff(m4+1,m3+1,m2+1,m1+1)= IntegralCoeff(1,m4+1,m3+1,m2+1)*1/sqrt(l0(2)-1+1)/sqrt(2*(l0(1)-1)+(l0(2)-1)+1);
                end
                l0(m1)=l0(m1)+1;
                %IntegralCoeff(m4+1,m3+1,m2+1,m1+1)=IntegralCoeff4(m4,m3,m2,m1);
                IntegralCoeffdt(m4+1,m3+1,m2+1,m1+1)=IntegralCoeff(m4+1,m3+1,m2+1,m1+1)* ...
                    1/(l0(1)-1+1)*(1-sqrt(l0(2)-1)/sqrt(2*(l0(1)-1)+(l0(2)-1)+2));
                IntegralCoeffdz(m4+1,m3+1,m2+1,m1+1)= IntegralCoeff(m4+1,m3+1,m2+1,m1+1)* ...
                    1/sqrt(l0(2)-1+1)/sqrt(2*(l0(1)-1)+(l0(2)-1)+1);
            end
        end
    end
end
end

function [XDensity,IndexOut,IndexMax] = MakeDensityFromSimulation_Infiniti(X,Paths,BinSize,MaxCutOff )
%Processes monte carlo paths to return a series Xdensity as a function of IndexOut. IndexMax is the maximum value of index.
% 
Xmin=0;
Xmax=0;
for p=1:Paths
if(X(p)>MaxCutOff)
X(p)=MaxCutOff;
end
if(Xmin>real(X(p)))
Xmin=real(X(p));
end
if(Xmax<real(X(p)))
Xmax=real(X(p));
end
end


IndexMax=floor((Xmax-Xmin)/BinSize+.5)+1
XDensity(1:IndexMax)=0.0;

for p=1:Paths
index=real(floor(real(X(p)-Xmin)/BinSize+.5)+1);
if(real(index)<1)
index=1;
end
if(real(index)>IndexMax)
index=IndexMax;
end

XDensity(index)=XDensity(index)+1.0/Paths/BinSize;

end

IndexOut(1:IndexMax)=Xmin+(0:(IndexMax-1))*BinSize;

end

A few days after I wrote the above post, I was forcefully detained again for several weeks since I had started doing good research. During past twenty two years, I have been detained forcefully more than twenty times. Whenever my research or work progresses, I am detained again and again.
Here in post 775, written on Sun Nov 04, 2018 7:48 pm, I distributed the programs ItoTaylorCoeffsNew and ComputeIntegralCoeffs again. Here is the post

Sorry, I missed many supporting matlab files for the program in post 773. I am posting them here.
Code: Select all

function [Y] = ItoTaylorCoeffsNew(alpha,beta1,beta2,gamma)

%In the coefficient calculation program which calculates Y(l1,l2,l3,l4), 
%I have used four levels of looping each for relevant expansion order. 
%The first loop takes four values and second loop takes 16 values and 
%third loop takes 64 values and so on. And then each coefficient 
%term can be individually calculated while carefully accounting 
%for path dependence. 
%So for example in a nested loop structure 
%m1= 1:mDim
%  m2=1:mDim
%  m3=1:mDim
%    l(m1)=l(m1)+1;
%    l(m2)=l(m2)+1;
%    l(m3)=l(m3)+1;
%in the above looping loop takes values from one to four with one 
%indicating the first drift term, two indicating the second drift term 
%and three indicating quadratic variation term and 
%four indicating the volatility term. And with this looping structure 
%we can So in the above looping m1=1 would mean that all terms are 
%descendent of first drift term and m2=4 would mean that all terms are 
%descendent of first drift term on first expansion order and descendent 
%of volatility term on the second order and so we can keep track of path 
%dependence perfectly. 
%And on each level, we individually calculate the descendent terms. While 
%keeping track of path dependence and calculating the coefficients with 
%careful path dependence consideration, we update the appropriate element 
%in our polynomial like expansion coefficient array
%explaining the part of code
%m1= 1:mDim
% m2=1:mDim
%  m3=1:mDim
%    l(m1)=l(m1)+1;
%    l(m2)=l(m2)+1;
%    l(m3)=l(m3)+1;
%Y(l(1)+1,l(2),l(3),l(4))=Y(l(1)+1,l(2),l(3),l(4))+Coeff1st*IntegralCoeff(1,1,1,2);
%Here l(1) denotes l1 but written as l(1) so it can be conveniently 
%updated with the loop variable when the loop variable takes value one 
%indicating first drift term . And l(2) could be conveniently updated when 
%the loop variable takes value equal to two indicating second 
%drift term and so on.
%Here is the part of code snippet for that
%for m1=1:mDim
%    l(1)=1;
%    l(2)=1;
%    l(3)=1;
%    l(4)=1;
%    l(m1)=l(m1)+1;
%CoeffDX1 = alpha + (l(1)-1) *beta1 + (l(2)-1) *beta2 + (l(3)-1) *2*gamma + (l(4)-1)*gamma - (l(1)-1) - (l(2)-1) - 2*(l(3)-1) - (l(4)-1);
%    CoeffDX2 = CoeffDX1 - 1;
%    ArrIndex0=m1;
%    ArrIndex=(m1-1)*mDim;
%    Coeff1st=Y1(ArrIndex0)*CoeffDX1;
%    Coeff2nd=Y1(ArrIndex0)*.5*CoeffDX1*CoeffDX2;
%    Y2(1+ArrIndex)=Coeff1st;
%    Y(l(1)+1,l(2),l(3),l(4))=Y(l(1)+1,l(2),l(3),l(4))+Coeff1st*IntegralCoeff(1,1,2,n1(m1));
%    Y2(2+ArrIndex)=Coeff1st;
%    Y(l(1),l(2)+1,l(3),l(4))=Y(l(1),l(2)+1,l(3),l(4))+Coeff1st*IntegralCoeff(1,1,2,n1(m1));
%    Y2(3+ArrIndex)=Coeff2nd;
%    Y(l(1),l(2),l(3)+1,l(4))=Y(l(1),l(2),l(3)+1,l(4))+Coeff2nd*IntegralCoeff(1,1,2,n1(m1));
%    Y2(4+ArrIndex)=Coeff1st;
%    Y(l(1),l(2),l(3),l(4)+1)=Y(l(1),l(2),l(3),l(4)+1)+Coeff1st*IntegralCoeff(1,1,3,n1(m1));
 
%The first four lines update the array indices according to the parent term. 
%And then CoeffDX1 and CoeffDX2 are calculated according to algebraic exponents on parent terms.    
%ArrIndex0=m1; calculates the array index of the parent term 
%And ArrIndex=(m1-1)*mDim; calculates the array index of the descendent terms
%And coefficient of the drift and volatility descendent terms is 
%calculated by multiplying the coefficient of the parent term by 
%Coeff1st=Y1(ArrIndex0)*CoeffDX1;
%And coefficient of the quadratic variation descendent terms is 
%calculated by multiplying the coefficient of the parent term by 
%Coeff2nd=Y1(ArrIndex0)*.5*CoeffDX1*CoeffDX2;
%And then each of the four descendent terms are updated with Coeff1st 
%if they are drift or volatility descendent terms or Coeff2nd if 
%they are quadratic variation descendent terms.
%Here Y1 indicates the temporary coefficient array with parent terms on 
%first level. And Y2 denotes temporary coefficient array with parent terms 
%on second level and Y3 indicates temporary coefficient array with parent terms on third level.
[IntegralCoeff,IntegralCoeffdt,IntegralCoeffdz] = ComputeIntegralCoeffs();
n1(1)=2;
n1(2)=2;
n1(3)=2;
n1(4)=3;
%n1(1), n1(2), n1(3) are drift and quadratic variation term variables 
%and take a value equal to 2 indicating a dt integral.
%n1(4) is volatility term variable and indicates a dz-integral by taking a
%value of 3.
mDim=4; % four descendent terms in each expansion
Y(1:5,1:5,1:5,1:5)=0;
Y1(1:mDim)=0;
Y2(1:mDim*mDim)=0;
Y3(1:mDim*mDim*mDim)=0;

%First Ito-hermite expansion level starts here. No loops but four
%descendent terms.
l(1)=1;
l(2)=1;
l(3)=1;
l(4)=1;
CoeffDX1 = alpha;
CoeffDX2 = CoeffDX1 - 1;
Coeff1st=CoeffDX1;
Coeff2nd=.5*CoeffDX1*CoeffDX2;
Y1(1)=Coeff1st;
Y(l(1)+1,l(2),l(3),l(4))=Y(l(1)+1,l(2),l(3),l(4))+Coeff1st*IntegralCoeff(1,1,1,2);
Y1(2)=Coeff1st;
Y(l(1),l(2)+1,l(3),l(4))=Y(l(1),l(2)+1,l(3),l(4))+Coeff1st*IntegralCoeff(1,1,1,2);
Y1(3)=Coeff2nd;
Y(l(1),l(2),l(3)+1,l(4))=Y(l(1),l(2),l(3)+1,l(4))+Coeff2nd*IntegralCoeff(1,1,1,2);
Y1(4)=Coeff1st;
Y(l(1),l(2),l(3),l(4)+1)=Y(l(1),l(2),l(3),l(4)+1)+Coeff1st*IntegralCoeff(1,1,1,3);
%Second Ito-hermite expansion level starts. It has a loop over four parent
%terms and there are four descendent terms for each parent term.
%The coefficient terms are then lumped in a polynomial-like expansion
%array of coefficients.
for m1=1:mDim
    l(1)=1;
    l(2)=1;
    l(3)=1;
    l(4)=1;
    l(m1)=l(m1)+1;
    CoeffDX1 = alpha + (l(1)-1) *beta1 + (l(2)-1) *beta2 + (l(3)-1) *2*gamma + (l(4)-1)*gamma ...
                        - (l(1)-1) - (l(2)-1) - 2*(l(3)-1) - (l(4)-1);
    CoeffDX2 = CoeffDX1 - 1;
    ArrIndex0=m1;
    ArrIndex=(m1-1)*mDim;
    Coeff1st=Y1(ArrIndex0)*CoeffDX1;
    Coeff2nd=Y1(ArrIndex0)*.5*CoeffDX1*CoeffDX2;
    Y2(1+ArrIndex)=Coeff1st;
    Y(l(1)+1,l(2),l(3),l(4))=Y(l(1)+1,l(2),l(3),l(4))+Coeff1st*IntegralCoeff(1,1,2,n1(m1));
    Y2(2+ArrIndex)=Coeff1st;
    Y(l(1),l(2)+1,l(3),l(4))=Y(l(1),l(2)+1,l(3),l(4))+Coeff1st*IntegralCoeff(1,1,2,n1(m1));
    Y2(3+ArrIndex)=Coeff2nd;
    Y(l(1),l(2),l(3)+1,l(4))=Y(l(1),l(2),l(3)+1,l(4))+Coeff2nd*IntegralCoeff(1,1,2,n1(m1));
    Y2(4+ArrIndex)=Coeff1st;
    Y(l(1),l(2),l(3),l(4)+1)=Y(l(1),l(2),l(3),l(4)+1)+Coeff1st*IntegralCoeff(1,1,3,n1(m1));
    %Third Ito-hermite expansion level starts and it is a nested loop with
    %a total of sixteen parents and each parent takes four descendent
    %terms.
    for m2=1:mDim
        l(1)=1;
        l(2)=1;
        l(3)=1;
        l(4)=1;
        l(m1)=l(m1)+1;
        l(m2)=l(m2)+1;
        CoeffDX1 = alpha + (l(1)-1) *beta1 + (l(2)-1) *beta2 + (l(3)-1) *2*gamma + (l(4)-1)*gamma ...
                        - (l(1)-1) - (l(2)-1) - 2*(l(3)-1) - (l(4)-1);
        CoeffDX2=CoeffDX1-1;            
        ArrIndex0=(m1-1)*mDim+m2;
        ArrIndex=((m1-1)*mDim+(m2-1))*mDim;
        Coeff1st=Y2(ArrIndex0)*CoeffDX1;
        Coeff2nd=Y2(ArrIndex0)*.5*CoeffDX1*CoeffDX2;
        Y3(1+ArrIndex)=Coeff1st;
        Y(l(1)+1,l(2),l(3),l(4))=Y(l(1)+1,l(2),l(3),l(4))+Coeff1st*IntegralCoeff(1,2,n1(m2),n1(m1));
        Y3(2+ArrIndex)=Coeff1st;
        Y(l(1),l(2)+1,l(3),l(4))=Y(l(1),l(2)+1,l(3),l(4))+Coeff1st*IntegralCoeff(1,2,n1(m2),n1(m1));
        Y3(3+ArrIndex)=Coeff2nd;
        Y(l(1),l(2),l(3)+1,l(4))=Y(l(1),l(2),l(3)+1,l(4))+Coeff2nd*IntegralCoeff(1,2,n1(m2),n1(m1));
        Y3(4+ArrIndex)=Coeff1st;
        Y(l(1),l(2),l(3),l(4)+1)=Y(l(1),l(2),l(3),l(4)+1)+Coeff1st*IntegralCoeff(1,3,n1(m2),n1(m1));
        %fourht Ito-hermite expansion level starts and it is a triply-nested loop with
        %a total of sixteen parents and each parent takes four descendent
        %terms. We then lump the terms in a relatively sparse polynomial 
        %like expansion coefficient array that has smaller number of
        %non-zero terms.
        for m3=1:mDim
            l(1)=1;
            l(2)=1;
            l(3)=1;
            l(4)=1;
            l(m1)=l(m1)+1;
            l(m2)=l(m2)+1;
            l(m3)=l(m3)+1;
            CoeffDX1 = alpha + (l(1)-1) *beta1 + (l(2)-1) *beta2 + (l(3)-1) *2*gamma + (l(4)-1)*gamma ...
                        - (l(1)-1) - (l(2)-1) - 2*(l(3)-1) - (l(4)-1);
            CoeffDX2=CoeffDX1-1;
            ArrIndex0=((m1-1)*mDim+(m2-1))*mDim+m3;
            %ArrIndex=(((m1-1)*mDim+(m2-1))*mDim+(m3-1))*mDim;
            Coeff1st=Y3(ArrIndex0)*CoeffDX1;
            Coeff2nd=Y3(ArrIndex0)*.5*CoeffDX1*CoeffDX2;
            %Y4(1+ArrIndex)=Coeff1st;
            Y(l(1)+1,l(2),l(3),l(4))=Y(l(1)+1,l(2),l(3),l(4))+Coeff1st*IntegralCoeff(2,n1(m3),n1(m2),n1(m1));
            %Y4(2+ArrIndex)=Coeff1st;
            Y(l(1),l(2)+1,l(3),l(4))=Y(l(1),l(2)+1,l(3),l(4))+Coeff1st*IntegralCoeff(2,n1(m3),n1(m2),n1(m1));
            %Y4(3+ArrIndex)=Coeff2nd;
            Y(l(1),l(2),l(3)+1,l(4))=Y(l(1),l(2),l(3)+1,l(4))+Coeff2nd*IntegralCoeff(2,n1(m3),n1(m2),n1(m1));
            %Y4(4+ArrIndex)=Coeff1st;
            Y(l(1),l(2),l(3),l(4)+1)=Y(l(1),l(2),l(3),l(4)+1)+Coeff1st*IntegralCoeff(3,n1(m3),n1(m2),n1(m1));
         end
    end
end
        
        
       
end

Here is the second supporting file.

function [IntegralCoeff0,IntegralCoeffdt,IntegralCoeffdz] = ComputeIntegralCoeffs()

IntegralCoeff(1:3,1:3,1:3,1:3)=0;
IntegralCoeffdt(1:3,1:3,1:3,1:3)=0;
IntegralCoeffdz(1:3,1:3,1:3,1:3)=0;
IntegralCoeff0(1:3,1:3,1:3,1:3)=0;
IntegralCoeff0(1,1,1,2)=1;
IntegralCoeff0(1,1,1,3)=1;
IntegralCoeff0(1,1,2,2)=1/2;
IntegralCoeff0(1,1,3,2)=1-1/sqrt(3);
IntegralCoeff0(1,1,2,3)=1/sqrt(3);
IntegralCoeff0(1,1,3,3)=1/2;
IntegralCoeff0(1,2,2,2)=1/6;
IntegralCoeff0(1,3,2,2)=(1-1/sqrt(3))*1/2*(1-1/sqrt(5));
IntegralCoeff0(1,2,3,2)=1/sqrt(3)/2*(1-1/sqrt(5));
IntegralCoeff0(1,3,3,2)=1/2*(1-sqrt(2)/2);
IntegralCoeff0(1,2,2,3)=1/2*1/sqrt(5);
IntegralCoeff0(1,3,2,3)=(1-1/sqrt(3))*1/sqrt(2)*1/2;
IntegralCoeff0(1,2,3,3)=1/sqrt(3)/sqrt(2)*1/(2);
IntegralCoeff0(1,3,3,3)=1/6;
IntegralCoeff0(2,2,2,2)=1/24;
IntegralCoeff0(2,2,3,2)=1/2*1/sqrt(5)*1/3*(1-1/sqrt(7));
IntegralCoeff0(2,3,2,2)=1/sqrt(3)*1/2*(1-1/sqrt(5))*1/3*(1-1/sqrt(7));
IntegralCoeff0(3,2,2,2)=(1-1/sqrt(3))*1/2*(1-1/sqrt(5))*1/3*(1-1/sqrt(7));
IntegralCoeff0(3,3,2,2)=1/2*(1-sqrt(2)/2)*1/2*(1-sqrt(2)/sqrt(6));
IntegralCoeff0(2,3,3,2)=1/sqrt(3)*1/sqrt(2)*1/2*1/2*(1-sqrt(2)/sqrt(6));
IntegralCoeff0(3,2,3,2)=(1-1/sqrt(3))*1/sqrt(2)*1/2*1/2*(1-sqrt(2)/sqrt(6));
IntegralCoeff0(3,3,3,2)=1/6*(1-sqrt(3)/sqrt(5));
IntegralCoeff0(2,2,2,3)=1/6*1/sqrt(7);
IntegralCoeff0(2,2,3,3)=1/2*1/sqrt(5)*1/sqrt(2)*1/sqrt(6);
IntegralCoeff0(2,3,2,3)=1/sqrt(3)*1/2*(1-1/sqrt(5))*1/sqrt(2)*1/sqrt(6);
IntegralCoeff0(3,2,2,3)=(1-1/sqrt(3))*1/2*(1-1/sqrt(5))*1/sqrt(2)*1/sqrt(6);
IntegralCoeff0(3,3,2,3)=1/2*(1-sqrt(2)/2)*1/sqrt(3)*1/sqrt(5);
IntegralCoeff0(2,3,3,3)=1/sqrt(3)*1/sqrt(2)*1/sqrt(4)*1/sqrt(3)*(1/sqrt(5));
IntegralCoeff0(3,2,3,3)=(1-1/sqrt(3))*1/sqrt(2)*1/2*1/sqrt(3)*1/sqrt(5);
IntegralCoeff0(3,3,3,3)=1/24;

%Can also be calculated with algorithm below.
%here IntegralCoeffdt indicates the coefficients of a dt-integral.
%This dt-integral means that you are calculating the Ito-hermite expansion
%of  Integral X(t)^alpha dt with X(t) dynamics given by the SDE 
%here IntegralCoeffdz indicates the coefficients of a dz-integral.
%This dz-integral means that you are calculating the Ito-hermite expansion
%of  Integral X(t)^alpha dz(t) with X(t) dynamics given by the SDE
%IntegralCoeff is is associated with expansion of X(t)^alpha.
%IntegralCoeff below is not returned and IntegralCoeff0 manually calculated
%above is returned but both are the same.
l0(1:2)=1;
for m4=1:2
    l0(1)=1;
    l0(2)=1;
    
    %IntegralCoeff4(m4,1,1,1)=1;
    %IntegralCoeff4(m4,1,1,1)=1;
    %1 is added to m4 since index 1 stands for zero, 2 for one and three
    %for two.
    IntegralCoeff(1,1,1,m4+1)=1;
    l0(m4)=l0(m4)+1;
    IntegralCoeffdt(1,1,1,m4+1)=IntegralCoeff(1,1,1,m4+1)* ...
        1/(l0(1)-1+1)*(1-sqrt(l0(2)-1)/sqrt(2*(l0(1)-1)+(l0(2)-1)+2));
    IntegralCoeffdz(1,1,1,m4+1)= IntegralCoeff(1,1,1,m4+1)* ...
        1/sqrt(l0(2)-1+1)/sqrt(2*(l0(1)-1)+(l0(2)-1)+1);
    for m3=1:2
        l0(1)=1;
        l0(2)=1;
        l0(m4)=l0(m4)+1;
        
        if(m3==1)
            IntegralCoeff(1,1,m4+1,m3+1)=1/(l0(1)-1+1)*(1-sqrt(l0(2)-1)/sqrt(2*(l0(1)-1)+(l0(2)-1)+2));
        end
        if(m3==2)
            IntegralCoeff(1,1,m4+1,m3+1)= 1/sqrt(l0(2)-1+1)/sqrt(2*(l0(1)-1)+(l0(2)-1)+1);
        end
        l0(m3)=l0(m3)+1;
        %IntegralCoeff(1,1,m4+1,m3+1)=IntegralCoeff4(m4,m3,1,1);
        IntegralCoeffdt(1,1,m4+1,m3+1)=IntegralCoeff(1,1,m4+1,m3+1)* ...
            1/(l0(1)-1+1)*(1-sqrt(l0(2)-1)/sqrt(2*(l0(1)-1)+(l0(2)-1)+2));
        IntegralCoeffdz(1,1,m4+1,m3+1)= IntegralCoeff(1,1,m4+1,m3+1)* ...
            1/sqrt(l0(2)-1+1)/sqrt(2*(l0(1)-1)+(l0(2)-1)+1);
        for m2=1:2
            l0(1)=1;
            l0(2)=1;
            l0(m4)=l0(m4)+1;
            l0(m3)=l0(m3)+1;
            
            if(m2==1)
                IntegralCoeff(1,m4+1,m3+1,m2+1)=IntegralCoeff(1,1,m4+1,m3+1)*1/(l0(1)-1+1)*(1-sqrt(l0(2)-1)/sqrt(2*(l0(1)-1)+(l0(2)-1)+2));
            end
            if(m2==2)
                IntegralCoeff(1,m4+1,m3+1,m2+1)= IntegralCoeff(1,1,m4+1,m3+1)*1/sqrt(l0(2)-1+1)/sqrt(2*(l0(1)-1)+(l0(2)-1)+1);
            end
            l0(m2)=l0(m2)+1;
            %IntegralCoeff(1,m4+1,m3+1,m2+1)=IntegralCoeff4(m4,m3,m2,1);
            IntegralCoeffdt(1,m4+1,m3+1,m2+1)=IntegralCoeff(1,m4+1,m3+1,m2+1)* ...
                1/(l0(1)-1+1)*(1-sqrt(l0(2)-1)/sqrt(2*(l0(1)-1)+(l0(2)-1)+2));
            IntegralCoeffdz(1,m4+1,m3+1,m2+1)= IntegralCoeff(1,m4+1,m3+1,m2+1)* ...
                1/sqrt(l0(2)-1+1)/sqrt(2*(l0(1)-1)+(l0(2)-1)+1);
            for m1=1:2
                l0(1)=1;
                l0(2)=1;
                l0(m4)=l0(m4)+1;
                l0(m3)=l0(m3)+1;
                l0(m2)=l0(m2)+1;
                if(m1==1)
                    IntegralCoeff(m4+1,m3+1,m2+1,m1+1)=IntegralCoeff(1,m4+1,m3+1,m2+1)*1/(l0(1)-1+1)*(1-sqrt(l0(2)-1)/sqrt(2*(l0(1)-1)+(l0(2)-1)+2));
                end
                if(m1==2)
                    IntegralCoeff(m4+1,m3+1,m2+1,m1+1)= IntegralCoeff(1,m4+1,m3+1,m2+1)*1/sqrt(l0(2)-1+1)/sqrt(2*(l0(1)-1)+(l0(2)-1)+1);
                end
                l0(m1)=l0(m1)+1;
                %IntegralCoeff(m4+1,m3+1,m2+1,m1+1)=IntegralCoeff4(m4,m3,m2,m1);
                IntegralCoeffdt(m4+1,m3+1,m2+1,m1+1)=IntegralCoeff(m4+1,m3+1,m2+1,m1+1)* ...
                    1/(l0(1)-1+1)*(1-sqrt(l0(2)-1)/sqrt(2*(l0(1)-1)+(l0(2)-1)+2));
                IntegralCoeffdz(m4+1,m3+1,m2+1,m1+1)= IntegralCoeff(m4+1,m3+1,m2+1,m1+1)* ...
                    1/sqrt(l0(2)-1+1)/sqrt(2*(l0(1)-1)+(l0(2)-1)+1);
            end
        end
    end
end
end

And here is the graphing from monte carlo file.

function [XDensity,IndexOut,IndexMax] = MakeDensityFromSimulation_Infiniti(X,Paths,BinSize,MaxCutOff )
%Processes monte carlo paths to return a series Xdensity as a function of IndexOut. IndexMax is the maximum value of index.
%

Xmin=0;
Xmax=0;
for p=1:Paths
if(X(p)>MaxCutOff)
X(p)=MaxCutOff;
end
if(X(p)<0)
X(p)=0;
end
if(Xmin>real(X(p)))
Xmin=real(X(p));
end
if(Xmax<real(X(p)))
Xmax=real(X(p));
end
end

IndexMax=floor((Xmax-Xmin)/BinSize+.5)+1
XDensity(1:IndexMax)=0.0;
for p=1:Paths
index=real(floor(real(X(p)-Xmin)/BinSize+.5)+1);
if(real(index)<1)
index=1;
end
if(real(index)>IndexMax)
index=IndexMax;
end
XDensity(index)=XDensity(index)+1.0/Paths/BinSize;
end
IndexOut(1:IndexMax)=Xmin+(0:(IndexMax-1))*BinSize;
end

In post 782, I have distributed the I distributed the functions ItoTaylorCoeffsNew and ComputeIntegralCoeffs again. I am not copying the post here. These functions were distributed several times again in several posts during past one and a half year.
At the end of my post, I would like to mention for friends that Ito-Taylor expansion theory of SDEs was originally developed by Dr. Eckhard Platen and Dr. Wolfgang Wagner. Here are two first original papers in this area.
1. A Generalized Taylor Formula for Solutions of Stochastic Equations. By Eckhard Platen.
2. On a Taylor Formula for a Class of Ito Processes. By Eckhard Platen and Wolfgang Wagner.

Though it is popularly called Ito-Taylor expansion, it has also been called Platen and Wagner formula at many places in the literature acknowledging the original work of Platen and Wagner.

Amin · March 16th, 2020, 3:05 pm

I have been previously distributing the functions ItoTaylorCoeffsNew which calculates the coefficients for monte carlo simulation for each different stochastic differential equation with two drift terms and its sub-function ComputeIntegralCoeffs. These functions expand the coefficients for Ito-Taylor monte carlo to four successive derivatives. I have another higher order version of this function which I never distributed before. I am sharing it for friends in case they would like even higher order for special problems. This would work exactly with the older monte carlo program but you will have to specify in monte carlo part OrderA=8 or OrderM=8. Here is the program. I do hope this program reaches friends exactly and is not altered on internet as I have not distributed it before.

function [IntegralCoeff] = ComputeIntegralCoeffsOrder8()


IntegralCoeff(1:3,1:3,1:3,1:3,1:3,1:3,1:3,1:3)=0;
%IntegralCoeffdt(1:3,1:3,1:3,1:3)=0;
%IntegralCoeffdz(1:3,1:3,1:3,1:3)=0;

IntegralCoeff0(1:3,1:3,1:3,1:3)=0;

IntegralCoeff0(1,1,1,2)=1;
IntegralCoeff0(1,1,1,3)=1;

IntegralCoeff0(1,1,2,2)=1/2;
IntegralCoeff0(1,1,3,2)=1-1/sqrt(3);
IntegralCoeff0(1,1,2,3)=1/sqrt(3);
IntegralCoeff0(1,1,3,3)=1/2;

IntegralCoeff0(1,2,2,2)=1/6;
IntegralCoeff0(1,3,2,2)=(1-1/sqrt(3))*1/2*(1-1/sqrt(5));
IntegralCoeff0(1,2,3,2)=1/sqrt(3)/2*(1-1/sqrt(5));
IntegralCoeff0(1,3,3,2)=1/2*(1-sqrt(2)/2);
IntegralCoeff0(1,2,2,3)=1/2*1/sqrt(5);
IntegralCoeff0(1,3,2,3)=(1-1/sqrt(3))*1/sqrt(2)*1/2;
IntegralCoeff0(1,2,3,3)=1/sqrt(3)/sqrt(2)*1/(2);
IntegralCoeff0(1,3,3,3)=1/6;

IntegralCoeff0(2,2,2,2)=1/24;
IntegralCoeff0(2,2,3,2)=1/2*1/sqrt(5)*1/3*(1-1/sqrt(7));
IntegralCoeff0(2,3,2,2)=1/sqrt(3)*1/2*(1-1/sqrt(5))*1/3*(1-1/sqrt(7));
IntegralCoeff0(3,2,2,2)=(1-1/sqrt(3))*1/2*(1-1/sqrt(5))*1/3*(1-1/sqrt(7));
IntegralCoeff0(3,3,2,2)=1/2*(1-sqrt(2)/2)*1/2*(1-sqrt(2)/sqrt(6));
IntegralCoeff0(2,3,3,2)=1/sqrt(3)*1/sqrt(2)*1/2*1/2*(1-sqrt(2)/sqrt(6));
IntegralCoeff0(3,2,3,2)=(1-1/sqrt(3))*1/sqrt(2)*1/2*1/2*(1-sqrt(2)/sqrt(6));
IntegralCoeff0(3,3,3,2)=1/6*(1-sqrt(3)/sqrt(5));

IntegralCoeff0(2,2,2,3)=1/6*1/sqrt(7);
IntegralCoeff0(2,2,3,3)=1/2*1/sqrt(5)*1/sqrt(2)*1/sqrt(6);
IntegralCoeff0(2,3,2,3)=1/sqrt(3)*1/2*(1-1/sqrt(5))*1/sqrt(2)*1/sqrt(6);
IntegralCoeff0(3,2,2,3)=(1-1/sqrt(3))*1/2*(1-1/sqrt(5))*1/sqrt(2)*1/sqrt(6);
IntegralCoeff0(3,3,2,3)=1/2*(1-sqrt(2)/2)*1/sqrt(3)*1/sqrt(5);
IntegralCoeff0(2,3,3,3)=1/sqrt(3)*1/sqrt(2)*1/sqrt(4)*1/sqrt(3)*(1/sqrt(5));
IntegralCoeff0(3,2,3,3)=(1-1/sqrt(3))*1/sqrt(2)*1/2*1/sqrt(3)*1/sqrt(5);
IntegralCoeff0(3,3,3,3)=1/24;


%Can also be calculated with algorithm below.
%here IntegralCoeffdt indicates the coefficients of a dt-integral.
%This dt-integral means that you are calculating the Ito-hermite expansion
%of  Integral X(t)^alpha dt with X(t) dynamics given by the SDE 
%here IntegralCoeffdz indicates the coefficients of a dz-integral.
%This dz-integral means that you are calculating the Ito-hermite expansion
%of  Integral X(t)^alpha dz(t) with X(t) dynamics given by the SDE 

%IntegralCoeff is is associated with expansion of X(t)^alpha.
%IntegralCoeff below is not returned and IntegralCoeff0 manually calculated
%above is returned but both are the same.

l0(1:2)=1;

for m8=1:2
    l0(1)=1;
    l0(2)=1;
    
    %IntegralCoeff4(m4,1,1,1)=1;
    %IntegralCoeff4(m4,1,1,1)=1;
    %1 is added to m8 since index 1 stands for zero, 2 for one and three
    %for two.
    IntegralCoeff(1,1,1,1,1,1,1,m8+1)=1;
    for m7=1:2
        l0(1)=1;
        l0(2)=1;
        l0(m8)=l0(m8)+1;
        
        if(m7==1)
            IntegralCoeff(1,1,1,1,1,1,m8+1,m7+1)=1/(l0(1)-1+1)*(1-sqrt(l0(2)-1)/sqrt(2*(l0(1)-1)+(l0(2)-1)+2));
        end
        if(m7==2)
            IntegralCoeff(1,1,1,1,1,1,m8+1,m7+1)= 1/sqrt(l0(2)-1+1)/sqrt(2*(l0(1)-1)+(l0(2)-1)+1);
        end
        for m6=1:2
            l0(1)=1;
            l0(2)=1;
            l0(m8)=l0(m8)+1;
            l0(m7)=l0(m7)+1;
            
            if(m6==1)
                IntegralCoeff(1,1,1,1,1,m8+1,m7+1,m6+1)=IntegralCoeff(1,1,1,1,1,1,m8+1,m7+1)*1/(l0(1)-1+1)*(1-sqrt(l0(2)-1)/sqrt(2*(l0(1)-1)+(l0(2)-1)+2));
            end
            if(m6==2)
                IntegralCoeff(1,1,1,1,1,m8+1,m7+1,m6+1)=IntegralCoeff(1,1,1,1,1,1,m8+1,m7+1)*1/sqrt(l0(2)-1+1)/sqrt(2*(l0(1)-1)+(l0(2)-1)+1);
            end

            for m5=1:2
                l0(1)=1;
                l0(2)=1;
                l0(m8)=l0(m8)+1;
                l0(m7)=l0(m7)+1;
                l0(m6)=l0(m6)+1;
                if(m5==1)
                    IntegralCoeff(1,1,1,1,m8+1,m7+1,m6+1,m5+1)=IntegralCoeff(1,1,1,1,1,m8+1,m7+1,m6+1)*1/(l0(1)-1+1)*(1-sqrt(l0(2)-1)/sqrt(2*(l0(1)-1)+(l0(2)-1)+2));
                end
                if(m5==2)
                    IntegralCoeff(1,1,1,1,m8+1,m7+1,m6+1,m5+1)=IntegralCoeff(1,1,1,1,1,m8+1,m7+1,m6+1)*1/sqrt(l0(2)-1+1)/sqrt(2*(l0(1)-1)+(l0(2)-1)+1);
                end
                for m4=1:2
                    l0(1)=1;
                    l0(2)=1;
                    l0(m8)=l0(m8)+1;
                    l0(m7)=l0(m7)+1;
                    l0(m6)=l0(m6)+1;
                    l0(m5)=l0(m5)+1;
                    if(m4==1)
                        IntegralCoeff(1,1,1,m8+1,m7+1,m6+1,m5+1,m4+1)=IntegralCoeff(1,1,1,1,m8+1,m7+1,m6+1,m5+1)*1/(l0(1)-1+1)*(1-sqrt(l0(2)-1)/sqrt(2*(l0(1)-1)+(l0(2)-1)+2));
                    end
                    if(m4==2)
                        IntegralCoeff(1,1,1,m8+1,m7+1,m6+1,m5+1,m4+1)=IntegralCoeff(1,1,1,1,m8+1,m7+1,m6+1,m5+1)*1/sqrt(l0(2)-1+1)/sqrt(2*(l0(1)-1)+(l0(2)-1)+1);
                    end
                    for m3=1:2
                        l0(1)=1;
                        l0(2)=1;
                        l0(m8)=l0(m8)+1;
                        l0(m7)=l0(m7)+1;
                        l0(m6)=l0(m6)+1;
                        l0(m5)=l0(m5)+1;
                        l0(m4)=l0(m4)+1;
                        if(m3==1)
                            IntegralCoeff(1,1,m8+1,m7+1,m6+1,m5+1,m4+1,m3+1)=IntegralCoeff(1,1,1,m8+1,m7+1,m6+1,m5+1,m4+1)*1/(l0(1)-1+1)*(1-sqrt(l0(2)-1)/sqrt(2*(l0(1)-1)+(l0(2)-1)+2));
                        end
                        if(m3==2)
                            IntegralCoeff(1,1,m8+1,m7+1,m6+1,m5+1,m4+1,m3+1)=IntegralCoeff(1,1,1,m8+1,m7+1,m6+1,m5+1,m4+1)*1/sqrt(l0(2)-1+1)/sqrt(2*(l0(1)-1)+(l0(2)-1)+1);
                        end
                        for m2=1:2
                            l0(1)=1;
                            l0(2)=1;
                            l0(m8)=l0(m8)+1;
                            l0(m7)=l0(m7)+1;
                            l0(m6)=l0(m6)+1;
                            l0(m5)=l0(m5)+1;
                            l0(m4)=l0(m4)+1;
                            l0(m3)=l0(m3)+1;
                            if(m2==1)
                                IntegralCoeff(1,m8+1,m7+1,m6+1,m5+1,m4+1,m3+1,m2+1)=IntegralCoeff(1,1,m8+1,m7+1,m6+1,m5+1,m4+1,m3+1)*1/(l0(1)-1+1)*(1-sqrt(l0(2)-1)/sqrt(2*(l0(1)-1)+(l0(2)-1)+2));
                            end
                            if(m2==2)
                                IntegralCoeff(1,m8+1,m7+1,m6+1,m5+1,m4+1,m3+1,m2+1)=IntegralCoeff(1,1,m8+1,m7+1,m6+1,m5+1,m4+1,m3+1)*1/sqrt(l0(2)-1+1)/sqrt(2*(l0(1)-1)+(l0(2)-1)+1);
                            end
                            for m1=1:2
                                l0(1)=1;
                                l0(2)=1;
                                l0(m8)=l0(m8)+1;
                                l0(m7)=l0(m7)+1;
                                l0(m6)=l0(m6)+1;
                                l0(m5)=l0(m5)+1;
                                l0(m4)=l0(m4)+1;
                                l0(m3)=l0(m3)+1;
                                l0(m2)=l0(m2)+1;
                                if(m1==1)
                                    IntegralCoeff(m8+1,m7+1,m6+1,m5+1,m4+1,m3+1,m2+1,m1+1)=IntegralCoeff(1,m8+1,m7+1,m6+1,m5+1,m4+1,m3+1,m2+1)*1/(l0(1)-1+1)*(1-sqrt(l0(2)-1)/sqrt(2*(l0(1)-1)+(l0(2)-1)+2));
                                end
                                if(m1==2)
                                    IntegralCoeff(m8+1,m7+1,m6+1,m5+1,m4+1,m3+1,m2+1,m1+1)=IntegralCoeff(1,m8+1,m7+1,m6+1,m5+1,m4+1,m3+1,m2+1)*1/sqrt(l0(2)-1+1)/sqrt(2*(l0(1)-1)+(l0(2)-1)+1);
                                end
                            end
                        end
                    end
                end
            end

        end
    end
end
end

Here is the second function which computes the eighth order coefficients.

function [Y] = ItoTaylorCoeffsNewOrder8(alpha,beta1,beta2,gamma)


%In the coefficient calculation program which calculates Y(l1,l2,l3,l4), 
%I have used four levels of looping each for relevant expansion order. 
%The first loop takes four values and second loop takes 16 values and 
%third loop takes 64 values and so on. And then each coefficient 
%term can be individually calculated while carefully accounting 
%for path dependence. 
%So for example in a nested loop structure 
%m1= 1:mDim
% 	m2=1:mDim
%		m3=1:mDim
%    l(m1)=l(m1)+1;
%    l(m2)=l(m2)+1;
%    l(m3)=l(m3)+1;

%in the above looping loop takes values from one to four with one 
%indicating the first drift term, two indicating the second drift term 
%and three indicating quadratic variation term and 
%four indicating the volatility term. And with this looping structure 
%we can So in the above looping m1=1 would mean that all terms are 
%descendent of first drift term and m2=4 would mean that all terms are 
%descendent of first drift term on first expansion order and descendent 
%of volatility term on the second order and so we can keep track of path 
%dependence perfectly. 
%And on each level, we individually calculate the descendent terms. While 
%keeping track of path dependence and calculating the coefficients with 
%careful path dependence consideration, we update the appropriate element 
%in our polynomial like expansion coefficient array 

%explaining the part of code
%m1= 1:mDim
%	m2=1:mDim
%		m3=1:mDim
%    l(m1)=l(m1)+1;
%    l(m2)=l(m2)+1;
%    l(m3)=l(m3)+1;
%Y(l(1)+1,l(2),l(3),l(4))=Y(l(1)+1,l(2),l(3),l(4))+Coeff1st*IntegralCoeff(1,1,1,2);

%Here l(1) denotes l1 but written as l(1) so it can be conveniently 
%updated with the loop variable when the loop variable takes value one 
%indicating first drift term . And l(2) could be conveniently updated when 
%the loop variable takes value equal to two indicating second 
%drift term and so on.
%Here is the part of code snippet for that

%for m1=1:mDim
%    l(1)=1;
%    l(2)=1;
%    l(3)=1;
%    l(4)=1;
%    l(m1)=l(m1)+1;
%CoeffDX1 = alpha + (l(1)-1) *beta1 + (l(2)-1) *beta2 + (l(3)-1) *2*gamma + (l(4)-1)*gamma - (l(1)-1) - (l(2)-1) - 2*(l(3)-1) - (l(4)-1);
%    CoeffDX2 = CoeffDX1 - 1;
%    ArrIndex0=m1;
%    ArrIndex=(m1-1)*mDim;
%    Coeff1st=Y1(ArrIndex0)*CoeffDX1;
%    Coeff2nd=Y1(ArrIndex0)*.5*CoeffDX1*CoeffDX2;
%    Y2(1+ArrIndex)=Coeff1st;
%    Y(l(1)+1,l(2),l(3),l(4))=Y(l(1)+1,l(2),l(3),l(4))+Coeff1st*IntegralCoeff(1,1,2,n1(m1));
%    Y2(2+ArrIndex)=Coeff1st;
%    Y(l(1),l(2)+1,l(3),l(4))=Y(l(1),l(2)+1,l(3),l(4))+Coeff1st*IntegralCoeff(1,1,2,n1(m1));
%    Y2(3+ArrIndex)=Coeff2nd;
%    Y(l(1),l(2),l(3)+1,l(4))=Y(l(1),l(2),l(3)+1,l(4))+Coeff2nd*IntegralCoeff(1,1,2,n1(m1));
%    Y2(4+ArrIndex)=Coeff1st;
%    Y(l(1),l(2),l(3),l(4)+1)=Y(l(1),l(2),l(3),l(4)+1)+Coeff1st*IntegralCoeff(1,1,3,n1(m1));

 
%The first four lines update the array indices according to the parent term. 
%And then CoeffDX1 and CoeffDX2 are calculated according to algebraic exponents on parent terms.    
%ArrIndex0=m1; calculates the array index of the parent term 
%And ArrIndex=(m1-1)*mDim; calculates the array index of the descendent terms
%And coefficient of the drift and volatility descendent terms is 
%calculated by multiplying the coefficient of the parent term by 
%Coeff1st=Y1(ArrIndex0)*CoeffDX1;
%And coefficient of the quadratic variation descendent terms is 
%calculated by multiplying the coefficient of the parent term by 
%Coeff2nd=Y1(ArrIndex0)*.5*CoeffDX1*CoeffDX2;
%And then each of the four descendent terms are updated with Coeff1st 
%if they are drift or volatility descendent terms or Coeff2nd if 
%they are quadratic variation descendent terms.
%Here Y1 indicates the temporary coefficient array with parent terms on 
%first level. And Y2 denotes temporary coefficient array with parent terms 
%on second level and Y3 indicates temporary coefficient array with parent terms on third level.

[IntegralCoeff] = ComputeIntegralCoeffsOrder8();

n1(1)=2;
n1(2)=2;
n1(3)=2;
n1(4)=3;
%n1(1), n1(2), n1(3) are drift and quadratic variation term variables 
%and take a value equal to 2 indicating a dt integral.
%n1(4) is volatility term variable and indicates a dz-integral by taking a
%value of 3. 

mDim=4; % four descendent terms in each expansion
Y(1:9,1:9,1:9,1:9)=0;
Y1(1:mDim)=0;
Y2(1:mDim*mDim)=0;
Y3(1:mDim*mDim*mDim)=0;
Y4(1:mDim*mDim*mDim*mDim)=0;
Y5(1:mDim*mDim*mDim*mDim*mDim)=0;
Y6(1:mDim*mDim*mDim*mDim*mDim*mDim)=0;
Y7(1:mDim*mDim*mDim*mDim*mDim*mDim*mDim)=0;


%First Ito-hermite expansion level starts here. No loops but four
%descendent terms.
l(1)=1;
l(2)=1;
l(3)=1;
l(4)=1;
CoeffDX1 = alpha;
CoeffDX2 = CoeffDX1 - 1;
Coeff1st=CoeffDX1;
Coeff2nd=.5*CoeffDX1*CoeffDX2;

Y1(1)=Coeff1st;
Y(l(1)+1,l(2),l(3),l(4))=Y(l(1)+1,l(2),l(3),l(4))+Coeff1st*IntegralCoeff(1,1,1,1,1,1,1,2);
Y1(2)=Coeff1st;
Y(l(1),l(2)+1,l(3),l(4))=Y(l(1),l(2)+1,l(3),l(4))+Coeff1st*IntegralCoeff(1,1,1,1,1,1,1,2);
Y1(3)=Coeff2nd;
Y(l(1),l(2),l(3)+1,l(4))=Y(l(1),l(2),l(3)+1,l(4))+Coeff2nd*IntegralCoeff(1,1,1,1,1,1,1,2);
Y1(4)=Coeff1st;
Y(l(1),l(2),l(3),l(4)+1)=Y(l(1),l(2),l(3),l(4)+1)+Coeff1st*IntegralCoeff(1,1,1,1,1,1,1,3);

%Second Ito-hermite expansion level starts. It has a loop over four parent
%terms and there are four descendent terms for each parent term.
%The coefficient terms are then lumped in a polynomial-like expansion
%array of coefficients.
for m1=1:mDim
    l(1)=1;
    l(2)=1;
    l(3)=1;
    l(4)=1;
    l(m1)=l(m1)+1;
    CoeffDX1 = alpha + (l(1)-1) *beta1 + (l(2)-1) *beta2 + (l(3)-1) *2*gamma + (l(4)-1)*gamma ...
                        - (l(1)-1) - (l(2)-1) - 2*(l(3)-1) - (l(4)-1);
    CoeffDX2 = CoeffDX1 - 1;
    ArrIndex0=m1;
    ArrIndex=(m1-1)*mDim;
    Coeff1st=Y1(ArrIndex0)*CoeffDX1;
    Coeff2nd=Y1(ArrIndex0)*.5*CoeffDX1*CoeffDX2;
    Y2(1+ArrIndex)=Coeff1st;
    Y(l(1)+1,l(2),l(3),l(4))=Y(l(1)+1,l(2),l(3),l(4))+Coeff1st*IntegralCoeff(1,1,1,1,1,1,2,n1(m1));
    Y2(2+ArrIndex)=Coeff1st;
    Y(l(1),l(2)+1,l(3),l(4))=Y(l(1),l(2)+1,l(3),l(4))+Coeff1st*IntegralCoeff(1,1,1,1,1,1,2,n1(m1));
    Y2(3+ArrIndex)=Coeff2nd;
    Y(l(1),l(2),l(3)+1,l(4))=Y(l(1),l(2),l(3)+1,l(4))+Coeff2nd*IntegralCoeff(1,1,1,1,1,1,2,n1(m1));
    Y2(4+ArrIndex)=Coeff1st;
    Y(l(1),l(2),l(3),l(4)+1)=Y(l(1),l(2),l(3),l(4)+1)+Coeff1st*IntegralCoeff(1,1,1,1,1,1,3,n1(m1));
    %Third Ito-hermite expansion level starts and it is a nested loop with
    %a total of sixteen parents and each parent takes four descendent
    %terms.
    for m2=1:mDim
        l(1)=1;
        l(2)=1;
        l(3)=1;
        l(4)=1;
        l(m1)=l(m1)+1;
        l(m2)=l(m2)+1;
        CoeffDX1 = alpha + (l(1)-1) *beta1 + (l(2)-1) *beta2 + (l(3)-1) *2*gamma + (l(4)-1)*gamma ...
                        - (l(1)-1) - (l(2)-1) - 2*(l(3)-1) - (l(4)-1);
        CoeffDX2=CoeffDX1-1;            
        ArrIndex0=(m1-1)*mDim+m2;
        ArrIndex=((m1-1)*mDim+(m2-1))*mDim;
        Coeff1st=Y2(ArrIndex0)*CoeffDX1;
        Coeff2nd=Y2(ArrIndex0)*.5*CoeffDX1*CoeffDX2;
        Y3(1+ArrIndex)=Coeff1st;
        Y(l(1)+1,l(2),l(3),l(4))=Y(l(1)+1,l(2),l(3),l(4))+Coeff1st*IntegralCoeff(1,1,1,1,1,2,n1(m2),n1(m1));
        Y3(2+ArrIndex)=Coeff1st;
        Y(l(1),l(2)+1,l(3),l(4))=Y(l(1),l(2)+1,l(3),l(4))+Coeff1st*IntegralCoeff(1,1,1,1,1,2,n1(m2),n1(m1));
        Y3(3+ArrIndex)=Coeff2nd;
        Y(l(1),l(2),l(3)+1,l(4))=Y(l(1),l(2),l(3)+1,l(4))+Coeff2nd*IntegralCoeff(1,1,1,1,1,2,n1(m2),n1(m1));
        Y3(4+ArrIndex)=Coeff1st;
        Y(l(1),l(2),l(3),l(4)+1)=Y(l(1),l(2),l(3),l(4)+1)+Coeff1st*IntegralCoeff(1,1,1,1,1,3,n1(m2),n1(m1));
        %fourht Ito-hermite expansion level starts and it is a triply-nested loop with
        %a total of sixteen parents and each parent takes four descendent
        %terms. We then lump the terms in a relatively sparse polynomial 
        %like expansion coefficient array that has smaller number of
        %non-zero terms.

        for m3=1:mDim
            l(1)=1;
            l(2)=1;
            l(3)=1;
            l(4)=1;
            l(m1)=l(m1)+1;
            l(m2)=l(m2)+1;
            l(m3)=l(m3)+1;
            CoeffDX1 = alpha + (l(1)-1) *beta1 + (l(2)-1) *beta2 + (l(3)-1) *2*gamma + (l(4)-1)*gamma ...
                        - (l(1)-1) - (l(2)-1) - 2*(l(3)-1) - (l(4)-1);
            CoeffDX2=CoeffDX1-1;
            ArrIndex0=((m1-1)*mDim+(m2-1))*mDim+m3;
            ArrIndex=(((m1-1)*mDim+(m2-1))*mDim+(m3-1))*mDim;
            Coeff1st=Y3(ArrIndex0)*CoeffDX1;
            Coeff2nd=Y3(ArrIndex0)*.5*CoeffDX1*CoeffDX2;
            Y4(1+ArrIndex)=Coeff1st;
            Y(l(1)+1,l(2),l(3),l(4))=Y(l(1)+1,l(2),l(3),l(4))+Coeff1st*IntegralCoeff(1,1,1,1,2,n1(m3),n1(m2),n1(m1));
            Y4(2+ArrIndex)=Coeff1st;
            Y(l(1),l(2)+1,l(3),l(4))=Y(l(1),l(2)+1,l(3),l(4))+Coeff1st*IntegralCoeff(1,1,1,1,2,n1(m3),n1(m2),n1(m1));
            Y4(3+ArrIndex)=Coeff2nd;
            Y(l(1),l(2),l(3)+1,l(4))=Y(l(1),l(2),l(3)+1,l(4))+Coeff2nd*IntegralCoeff(1,1,1,1,2,n1(m3),n1(m2),n1(m1));
            Y4(4+ArrIndex)=Coeff1st;
            Y(l(1),l(2),l(3),l(4)+1)=Y(l(1),l(2),l(3),l(4)+1)+Coeff1st*IntegralCoeff(1,1,1,1,3,n1(m3),n1(m2),n1(m1));
            for m4=1:mDim
                l(1)=1;
                l(2)=1;
                l(3)=1;
                l(4)=1;
                l(m1)=l(m1)+1;
                l(m2)=l(m2)+1;
                l(m3)=l(m3)+1;
                l(m4)=l(m4)+1;
                CoeffDX1 = alpha + (l(1)-1) *beta1 + (l(2)-1) *beta2 + (l(3)-1) *2*gamma + (l(4)-1)*gamma ...
                        - (l(1)-1) - (l(2)-1) - 2*(l(3)-1) - (l(4)-1);
                CoeffDX2=CoeffDX1-1;
                
                ArrIndex0=(((m1-1)*mDim+(m2-1))*mDim+(m3-1))*mDim+m4;
                ArrIndex=((((m1-1)*mDim+(m2-1))*mDim+(m3-1))*mDim+(m4-1))*mDim;
                
                Coeff1st=Y4(ArrIndex0)*CoeffDX1;
                Coeff2nd=Y4(ArrIndex0)*.5*CoeffDX1*CoeffDX2;
                Y5(1+ArrIndex)=Coeff1st;
                Y(l(1)+1,l(2),l(3),l(4))=Y(l(1)+1,l(2),l(3),l(4))+Coeff1st*IntegralCoeff(1,1,1,2,n1(m4),n1(m3),n1(m2),n1(m1));
                Y5(2+ArrIndex)=Coeff1st;
                Y(l(1),l(2)+1,l(3),l(4))=Y(l(1),l(2)+1,l(3),l(4))+Coeff1st*IntegralCoeff(1,1,1,2,n1(m4),n1(m3),n1(m2),n1(m1));
                Y5(3+ArrIndex)=Coeff2nd;
                Y(l(1),l(2),l(3)+1,l(4))=Y(l(1),l(2),l(3)+1,l(4))+Coeff2nd*IntegralCoeff(1,1,1,2,n1(m4),n1(m3),n1(m2),n1(m1));
                Y5(4+ArrIndex)=Coeff1st;
                Y(l(1),l(2),l(3),l(4)+1)=Y(l(1),l(2),l(3),l(4)+1)+Coeff1st*IntegralCoeff(1,1,1,3,n1(m4),n1(m3),n1(m2),n1(m1));
                for m5=1:mDim
                    l(1)=1;
                    l(2)=1;
                    l(3)=1;
                    l(4)=1;
                    l(m1)=l(m1)+1;
                    l(m2)=l(m2)+1;
                    l(m3)=l(m3)+1;
                    l(m4)=l(m4)+1;
                    l(m5)=l(m5)+1;
                    CoeffDX1 = alpha + (l(1)-1) *beta1 + (l(2)-1) *beta2 + (l(3)-1) *2*gamma + (l(4)-1)*gamma ...
                        - (l(1)-1) - (l(2)-1) - 2*(l(3)-1) - (l(4)-1);
                    CoeffDX2=CoeffDX1-1;
                
                    ArrIndex0=((((m1-1)*mDim+(m2-1))*mDim+(m3-1))*mDim+(m4-1))*mDim+m5;
                    ArrIndex=(((((m1-1)*mDim+(m2-1))*mDim+(m3-1))*mDim+(m4-1))*mDim+(m5-1))*mDim;
                
                    Coeff1st=Y5(ArrIndex0)*CoeffDX1;
                    Coeff2nd=Y5(ArrIndex0)*.5*CoeffDX1*CoeffDX2;
                    Y6(1+ArrIndex)=Coeff1st;
                    Y(l(1)+1,l(2),l(3),l(4))=Y(l(1)+1,l(2),l(3),l(4))+Coeff1st*IntegralCoeff(1,1,2,n1(m5),n1(m4),n1(m3),n1(m2),n1(m1));
                    Y6(2+ArrIndex)=Coeff1st;
                    Y(l(1),l(2)+1,l(3),l(4))=Y(l(1),l(2)+1,l(3),l(4))+Coeff1st*IntegralCoeff(1,1,2,n1(m5),n1(m4),n1(m3),n1(m2),n1(m1));
                    Y6(3+ArrIndex)=Coeff2nd;
                    Y(l(1),l(2),l(3)+1,l(4))=Y(l(1),l(2),l(3)+1,l(4))+Coeff2nd*IntegralCoeff(1,1,2,n1(m5),n1(m4),n1(m3),n1(m2),n1(m1));
                    Y6(4+ArrIndex)=Coeff1st;
                    Y(l(1),l(2),l(3),l(4)+1)=Y(l(1),l(2),l(3),l(4)+1)+Coeff1st*IntegralCoeff(1,1,3,n1(m5),n1(m4),n1(m3),n1(m2),n1(m1));
                    for m6=1:mDim
                        l(1)=1;
                        l(2)=1;
                        l(3)=1;
                        l(4)=1;
                        l(m1)=l(m1)+1;
                        l(m2)=l(m2)+1;
                        l(m3)=l(m3)+1;
                        l(m4)=l(m4)+1;
                        l(m5)=l(m5)+1;
                        l(m6)=l(m6)+1;
                        CoeffDX1 = alpha + (l(1)-1) *beta1 + (l(2)-1) *beta2 + (l(3)-1) *2*gamma + (l(4)-1)*gamma ...
                            - (l(1)-1) - (l(2)-1) - 2*(l(3)-1) - (l(4)-1);
                        CoeffDX2=CoeffDX1-1;
                
                        ArrIndex0=(((((m1-1)*mDim+(m2-1))*mDim+(m3-1))*mDim+(m4-1))*mDim+(m5-1))*mDim+m6;
                        ArrIndex=((((((m1-1)*mDim+(m2-1))*mDim+(m3-1))*mDim+(m4-1))*mDim+(m5-1))*mDim+(m6-1))*mDim;
                
                        Coeff1st=Y6(ArrIndex0)*CoeffDX1;
                        Coeff2nd=Y6(ArrIndex0)*.5*CoeffDX1*CoeffDX2;
                        Y7(1+ArrIndex)=Coeff1st;
                        Y(l(1)+1,l(2),l(3),l(4))=Y(l(1)+1,l(2),l(3),l(4))+Coeff1st*IntegralCoeff(1,2,n1(m6),n1(m5),n1(m4),n1(m3),n1(m2),n1(m1));
                        Y7(2+ArrIndex)=Coeff1st;
                        Y(l(1),l(2)+1,l(3),l(4))=Y(l(1),l(2)+1,l(3),l(4))+Coeff1st*IntegralCoeff(1,2,n1(m6),n1(m5),n1(m4),n1(m3),n1(m2),n1(m1));
                        Y7(3+ArrIndex)=Coeff2nd;
                        Y(l(1),l(2),l(3)+1,l(4))=Y(l(1),l(2),l(3)+1,l(4))+Coeff2nd*IntegralCoeff(1,2,n1(m6),n1(m5),n1(m4),n1(m3),n1(m2),n1(m1));
                        Y7(4+ArrIndex)=Coeff1st;
                        Y(l(1),l(2),l(3),l(4)+1)=Y(l(1),l(2),l(3),l(4)+1)+Coeff1st*IntegralCoeff(1,3,n1(m6),n1(m5),n1(m4),n1(m3),n1(m2),n1(m1));
                        for m7=1:mDim
                            l(1)=1;
                            l(2)=1;
                            l(3)=1;
                            l(4)=1;
                            l(m1)=l(m1)+1;
                            l(m2)=l(m2)+1;
                            l(m3)=l(m3)+1;
                            l(m4)=l(m4)+1;
                            l(m5)=l(m5)+1;
                            l(m6)=l(m6)+1;
                            l(m7)=l(m7)+1;
                            CoeffDX1 = alpha + (l(1)-1) *beta1 + (l(2)-1) *beta2 + (l(3)-1) *2*gamma + (l(4)-1)*gamma ...
                                - (l(1)-1) - (l(2)-1) - 2*(l(3)-1) - (l(4)-1);
                            CoeffDX2=CoeffDX1-1;
                
                            ArrIndex0=((((((m1-1)*mDim+(m2-1))*mDim+(m3-1))*mDim+(m4-1))*mDim+(m5-1))*mDim+(m6-1))*mDim+m7;
                            ArrIndex=(((((((m1-1)*mDim+(m2-1))*mDim+(m3-1))*mDim+(m4-1))*mDim+(m5-1))*mDim+(m6-1))*mDim+(m7-1))*mDim;
                
                            Coeff1st=Y7(ArrIndex0)*CoeffDX1;
                            Coeff2nd=Y7(ArrIndex0)*.5*CoeffDX1*CoeffDX2;
                            %Y8(1+ArrIndex)=Coeff1st;
                            Y(l(1)+1,l(2),l(3),l(4))=Y(l(1)+1,l(2),l(3),l(4))+Coeff1st*IntegralCoeff(2,n1(m7),n1(m6),n1(m5),n1(m4),n1(m3),n1(m2),n1(m1));
                            %Y8(2+ArrIndex)=Coeff1st;
                            Y(l(1),l(2)+1,l(3),l(4))=Y(l(1),l(2)+1,l(3),l(4))+Coeff1st*IntegralCoeff(2,n1(m7),n1(m6),n1(m5),n1(m4),n1(m3),n1(m2),n1(m1));
                            %Y8(3+ArrIndex)=Coeff2nd;
                            Y(l(1),l(2),l(3)+1,l(4))=Y(l(1),l(2),l(3)+1,l(4))+Coeff2nd*IntegralCoeff(2,n1(m7),n1(m6),n1(m5),n1(m4),n1(m3),n1(m2),n1(m1));
                            %Y8(4+ArrIndex)=Coeff1st;
                            Y(l(1),l(2),l(3),l(4)+1)=Y(l(1),l(2),l(3),l(4)+1)+Coeff1st*IntegralCoeff(3,n1(m7),n1(m6),n1(m5),n1(m4),n1(m3),n1(m2),n1(m1));
                        end
                    end
                end
            end

         end
    end
end
        
end

Amin · March 17th, 2020, 2:45 pm

Though I have worked previously with SDEs that have no explicit time dependence, the method is equally applicable to SDEs that have an explicit time dependence in the drift or volatility but the calculation of iterated integrals becomes relatively tedious sometimes though it can still be easily done. I will give an example of SDE with a time exponential in volatility term and show how to expand its solution to higher order. Here is an SDE with time exponential in volatility term

[$]dx(t)=\mu(x) dt + \exp(-\kappa t) \sigma(x) dz(t)[$]

This SDE can easily be expanded to 2nd order as

[$]x(t)=x_0 + \int_0^t  \mu(x_0) d \tau + \int_0^t  \exp(-\kappa \tau) \sigma(x_0) d z(\tau) [$]
[$]+ \int_0^t \int_0^{\tau} \frac{d\mu(x_0)}{dx} \mu(x_0) ds d \tau[$]
[$]+ \int_0^t \int_0^{\tau} \frac{d\mu(x_0)}{dx} \exp(-\kappa s) \sigma(x_0) dz(s) d \tau[$]
[$]+ \int_0^t \int_0^{\tau}  \exp(-\kappa \tau) \frac{d\sigma(x_0)}{dx} \mu(x_0) ds dz(\tau) [$]
[$]+ \int_0^t  \int_0^{\tau}  \exp(-\kappa \tau) \frac{d\sigma(x_0)}{dx}  \exp(-\kappa s) \sigma(x_0) dz(s) dz(\tau) [$]
[$]+.5 \int_0^t \int_0^{\tau} \frac{d^2\mu(x_0)}{dx^2} {(\exp(-\kappa s) \sigma(x_0))}^2 ds d \tau[$]
[$]+.5 \int_0^t \int_0^{\tau}  \exp(-\kappa \tau) \frac{d^2\sigma(x_0)}{dx^2} {(\exp(-\kappa s) \sigma(x_0))}^2 ds dz( \tau)[$]
+ higher order terms

If you want to know how I expanded into the above solution terms, you will have to look at my paper on ODEs here where I have used this technique as time dependence is a very regular feature in ODEs. Here is the paper: https://papers.ssrn.com/sol3/papers.cfm?abstract_id=2872598

if we take the constant values evaluated at initial point out of integrals then the equations become

[$]x(t)=x_0 +\mu(x_0) \int_0^t d \tau + \sigma(x_0) \int_0^t  \exp(-\kappa \tau) d z(\tau) [$]
[$]+\frac{d\mu(x_0)}{dx} \mu(x_0) \int_0^t \int_0^{\tau} ds d \tau[$]
[$]+\frac{d\mu(x_0)}{dx} \sigma(x_0) \int_0^t \int_0^{\tau} \exp(-\kappa s) dz(s) d \tau[$]
[$]+\frac{d\sigma(x_0)}{dx} \mu(x_0)  \int_0^t \int_0^{\tau}  \exp(-\kappa \tau) ds dz(\tau) [$]
[$]+\frac{d\sigma(x_0)}{dx} \sigma(x_0) \int_0^t  \int_0^{\tau}  \exp(-\kappa \tau) \exp(-\kappa s) dz(s) dz(\tau) [$]
[$]+.5 \frac{d^2\mu(x_0)}{dx^2} {\sigma(x_0)}^2 \int_0^t \int_0^{\tau} \exp(-2 \kappa s) ds d \tau[$]
[$]+.5 \frac{d^2\sigma(x_0)}{dx^2} {\sigma(x_0)}^2 \int_0^t \int_0^{\tau}  \exp(-\kappa \tau) \exp(-2 \kappa s) ds dz( \tau)[$]
+ higher order terms

So we are left with solving the stochastic integrals with time exponential terms. The integrals in first three terms are straightforward.
The integral in fourth term is not straightforward at all but we can still do it. Here is the integral
[$] \int_0^t \int_0^{\tau} \exp(-\kappa s) dz(s) d \tau[$]
To solve this integral, we will have to first convert the exponential into its Taylor series and then integrate each Taylor term one by one.
This can be written as
[$] \int_0^t \int_0^{\tau} dz(s) d \tau-\kappa \int_0^t \int_0^{\tau} s dz(s) d \tau+.5 {\kappa}^2 \int_0^t \int_0^{\tau} s^2 dz(s) d \tau[$]
[$]-1/6 {\kappa}^3 \int_0^t \int_0^{\tau} s^3 dz(s) d \tau +1/24 {\kappa}^4 \int_0^t \int_0^{\tau} s^4 dz(s) d \tau[$]
each of the above integrals can be solved by our hermite iterated integral update formulas given below

[$]\int_0^t s^m H_{n}(z(s)) dz(s)= \frac {t^m H_{n+1}(z(t))}{ \sqrt{(n+1)} \sqrt{(2m+n+1)}} [$]
[$]\int_0^t s^m H_{n}(z(s)) ds = \frac {t^{m+1} H_{n}(z(t))}{(m+1)} (1 - \frac {\sqrt{n}}{\sqrt{(2m+n+2)}}) [$]

Then we come to solution of fifth integral given below
[$]+\frac{d\sigma(x_0)}{dx} \mu(x_0)  \int_0^t \int_0^{\tau}  \exp(-\kappa \tau) ds dz(\tau) [$]

The above integral can easily be solved by solving the inner integral using calculus and then using Ito isometry for outer integral. The solutionof inner integral gives us [$]\tau \exp(-\kappa \tau)[$] and then we get the solution of variance from Ito-isometry as
[$]\int_0^t {\tau}^2 \exp(-2 \kappa \tau) d\tau= (1+\exp(-2 \kappa t) (-1 - 2 \kappa t (1+\kappa t)))/(4 {\kappa}^3)[$]
We could have also solved the above integral after converting exponential into taylor form and integrating using our iterated integral update formulas but that would probably be unnecessary here.
And the sixth integral is
[$]\int_0^t  \int_0^{\tau}  \exp(-\kappa \tau) \exp(-\kappa s) dz(s) dz(\tau) [$]

This integral can also be solved by repeated application of Ito-isometry just like we solved simple iterated stochastic integrals.
solution of variance from inner integral would give us
[$]\frac{1}{2 \kappa} (1-\exp(-2 \kappa \tau) ) \exp(-2 \kappa \tau)[$]
and its integral again would give us the variance
[$]\exp(-2 \kappa t) {Sinh(\kappa t)}^2/(2 {\kappa}^2)[$]

Seventh and eighth quadratic terms integrals can also be easily solved by the same method.

In a similar fashion, we can have an exponential or some other explicit time function also in the drift or both in volatility and drift and we could easily solve those stochastic differential equations
So we can basically solve many many other stochastic differential equations with explicit time dependence with this method but some stochastic integrals could only be solved if we can find a taylor series representation of the time dependent part of the term.

Amin · March 19th, 2020, 9:13 am

In post 892 above, I have explained my contribution to the simulation of Stochastic differential equations( and very briefly to ordinary differential equations) with timeline of different posts. I want to request friends to be kind and please give me credit for my work. Even if there is no proper formal paper, please cite the appropriate Wilmott posts with my name. The whole work has been floating informally on internet for a long time and I am sure several people might have rediscovered some of these things on their own after getting partial information from internet or other sources. If you know someone like that please inform them of my work and ask them to give me credit for my work. I absolutely welcome friends who would like to do or who would already have done more advanced work after this but I would still request them to give me credit for my research.
Since my formal mathematics background is rather limited, I would also request other intelligent friends to more rigorously formalize the ideas I used in the solution of Ordinary differential equations (using a method that is a cousin of Taylor series) and point out any further extensions and applications of the method. Can it somehow intelligently be applied to PDEs?
Also please tell your friends who work with ordinary differential equations about my paper on ODEs. Here is the link to my paper
https://papers.ssrn.com/sol3/papers.cfm?abstract_id=2872598

Amin · March 19th, 2020, 3:19 pm

Though I have worked previously with SDEs that have no explicit time dependence, the method is equally applicable to SDEs that have an explicit time dependence in the drift or volatility but the calculation of iterated integrals becomes relatively tedious sometimes though it can still be easily done. I will give an example of SDE with a time exponential in volatility term and show how to expand its solution to higher order. Here is an SDE with time exponential in volatility term

[$]dx(t)=\mu(x) dt + \exp(-\kappa t) \sigma(x) dz(t)[$]

This SDE can easily be expanded to 2nd order as

[$]x(t)=x_0 + \int_0^t  \mu(x_0) d \tau + \int_0^t  \exp(-\kappa \tau) \sigma(x_0) d z(\tau) [$]
[$]+ \int_0^t \int_0^{\tau} \frac{d\mu(x_0)}{dx} \mu(x_0) ds d \tau[$]
[$]+ \int_0^t \int_0^{\tau} \frac{d\mu(x_0)}{dx} \exp(-\kappa s) \sigma(x_0) dz(s) d \tau[$]
[$]+ \int_0^t \int_0^{\tau}  \exp(-\kappa \tau) \frac{d\sigma(x_0)}{dx} \mu(x_0) ds dz(\tau) [$]
[$]+ \int_0^t  \int_0^{\tau}  \exp(-\kappa \tau) \frac{d\sigma(x_0)}{dx}  \exp(-\kappa s) \sigma(x_0) dz(s) dz(\tau) [$]
[$]+.5 \int_0^t \int_0^{\tau} \frac{d^2\mu(x_0)}{dx^2} {(\exp(-\kappa s) \sigma(x_0))}^2 ds d \tau[$]
[$]+.5 \int_0^t \int_0^{\tau}  \exp(-\kappa \tau) \frac{d^2\sigma(x_0)}{dx^2} {(\exp(-\kappa s) \sigma(x_0))}^2 ds dz( \tau)[$]
+ higher order terms

If you want to know how I expanded into the above solution terms, you will have to look at my paper on ODEs here where I have used this technique as time dependence is a very regular feature in ODEs. Here is the paper: https://papers.ssrn.com/sol3/papers.cfm?abstract_id=2872598

if we take the constant values evaluated at initial point out of integrals then the equations become

[$]x(t)=x_0 +\mu(x_0) \int_0^t d \tau + \sigma(x_0) \int_0^t  \exp(-\kappa \tau) d z(\tau) [$]
[$]+\frac{d\mu(x_0)}{dx} \mu(x_0) \int_0^t \int_0^{\tau} ds d \tau[$]
[$]+\frac{d\mu(x_0)}{dx} \sigma(x_0) \int_0^t \int_0^{\tau} \exp(-\kappa s) dz(s) d \tau[$]
[$]+\frac{d\sigma(x_0)}{dx} \mu(x_0)  \int_0^t \int_0^{\tau}  \exp(-\kappa \tau) ds dz(\tau) [$]
[$]+\frac{d\sigma(x_0)}{dx} \sigma(x_0) \int_0^t  \int_0^{\tau}  \exp(-\kappa \tau) \exp(-\kappa s) dz(s) dz(\tau) [$]
[$]+.5 \frac{d^2\mu(x_0)}{dx^2} {\sigma(x_0)}^2 \int_0^t \int_0^{\tau} \exp(-2 \kappa s) ds d \tau[$]
[$]+.5 \frac{d^2\sigma(x_0)}{dx^2} {\sigma(x_0)}^2 \int_0^t \int_0^{\tau}  \exp(-\kappa \tau) \exp(-2 \kappa s) ds dz( \tau)[$]
+ higher order terms

So we are left with solving the stochastic integrals with time exponential terms. The integrals in first three terms are straightforward.
The integral in fourth term is not straightforward at all but we can still do it. Here is the integral
[$] \int_0^t \int_0^{\tau} \exp(-\kappa s) dz(s) d \tau[$]
To solve this integral, we will have to first convert the exponential into its Taylor series and then integrate each Taylor term one by one.
This can be written as
[$] \int_0^t \int_0^{\tau} dz(s) d \tau-\kappa \int_0^t \int_0^{\tau} s dz(s) d \tau+.5 {\kappa}^2 \int_0^t \int_0^{\tau} s^2 dz(s) d \tau[$]
[$]-1/6 {\kappa}^3 \int_0^t \int_0^{\tau} s^3 dz(s) d \tau +1/24 {\kappa}^4 \int_0^t \int_0^{\tau} s^4 dz(s) d \tau[$]
each of the above integrals can be solved by our hermite iterated integral update formulas given below

[$]\int_0^t s^m H_{n}(z(s)) dz(s)= \frac {t^m H_{n+1}(z(t))}{ \sqrt{(n+1)} \sqrt{(2m+n+1)}} [$]
[$]\int_0^t s^m H_{n}(z(s)) ds = \frac {t^{m+1} H_{n}(z(t))}{(m+1)} (1 - \frac {\sqrt{n}}{\sqrt{(2m+n+2)}}) [$]

Then we come to solution of fifth integral given below
[$]+\frac{d\sigma(x_0)}{dx} \mu(x_0)  \int_0^t \int_0^{\tau}  \exp(-\kappa \tau) ds dz(\tau) [$]

The above integral can easily be solved by solving the inner integral using calculus and then using Ito isometry for outer integral. The solutionof inner integral gives us [$]\tau \exp(-\kappa \tau)[$] and then we get the solution of variance from Ito-isometry as
[$]\int_0^t {\tau}^2 \exp(-2 \kappa \tau) d\tau= (1+\exp(-2 \kappa t) (-1 - 2 \kappa t (1+\kappa t)))/(4 {\kappa}^3)[$]
We could have also solved the above integral after converting exponential into taylor form and integrating using our iterated integral update formulas but that would probably be unnecessary here.
And the sixth integral is
[$]\int_0^t  \int_0^{\tau}  \exp(-\kappa \tau) \exp(-\kappa s) dz(s) dz(\tau) [$]

This integral can also be solved by repeated application of Ito-isometry just like we solved simple iterated stochastic integrals.
solution of variance from inner integral would give us
[$]\frac{1}{2 \kappa} (1-\exp(-2 \kappa \tau) ) \exp(-2 \kappa \tau)[$]
and its integral again would give us the variance
[$]\exp(-2 \kappa t) {Sinh(\kappa t)}^2/(2 {\kappa}^2)[$]

Seventh and eighth quadratic terms integrals can also be easily solved by the same method.

In a similar fashion, we can have an exponential or some other explicit time function also in the drift or both in volatility and drift and we could easily solve those stochastic differential equations
So we can basically solve many many other stochastic differential equations with explicit time dependence with this method but some stochastic integrals could only be solved if we can find a taylor series representation of the time dependent part of the term.

Here I copy this part from the above post

The integral in fourth term is not straightforward at all but we can still do it. Here is the integral
[$] \int_0^t \int_0^{\tau} \exp(-\kappa s) dz(s) d \tau[$]
To solve this integral, we will have to first convert the exponential into its Taylor series and then integrate each Taylor term one by one.
This can be written as
[$] \int_0^t \int_0^{\tau} dz(s) d \tau-\kappa \int_0^t \int_0^{\tau} s dz(s) d \tau+.5 {\kappa}^2 \int_0^t \int_0^{\tau} s^2 dz(s) d \tau[$]
[$]-1/6 {\kappa}^3 \int_0^t \int_0^{\tau} s^3 dz(s) d \tau +1/24 {\kappa}^4 \int_0^t \int_0^{\tau} s^4 dz(s) d \tau[$]
each of the above integrals can be solved by our hermite iterated integral update formulas given below

[$]\int_0^t s^m H_{n}(z(s)) dz(s)= \frac {t^m H_{n+1}(z(t))}{ \sqrt{(n+1)} \sqrt{(2m+n+1)}} [$]
[$]\int_0^t s^m H_{n}(z(s)) ds = \frac {t^{m+1} H_{n}(z(t))}{(m+1)} (1 - \frac {\sqrt{n}}{\sqrt{(2m+n+2)}}) [$]

So I am writing this post as it can be an interesting small project for a graduate student to find the solution of above integral by converting exponential into taylor series and then using hermite iterated integral update formulas and once the solution has been found, it would be interesting to see if there is some other analytic way to solve these integrals as a single term especially if the series answer could again be reconstructed in the form of an exponential. And if there is some analytic way to do that then it could probably be broadened/applied to time functions other than the exponentials.
Though not related to above integrals, Another very interesting research possibility could be if we could somehow take a different power of time other than the square-root in dz(t) in the SDE.

Amin · March 20th, 2020, 3:58 am

Since I think it would be nice and valuable, I have decided to write a very general program with explicitly time dependent coefficients. There is no conceptual problems in that and it is purely a work in developing a good algorithm with already defined concepts. Here is what a general SDE would look like in integral form

[$]x(T)=x_0 + g_1(T) \int_0^T f_1(t) \mu_1 x(t)^{\beta_1} dt + g_2(T) \int_0^T f_2(t) \mu_2 x(t)^{\beta_2} dt[$]
[$]+ g_3(T) \int_0^T f_3(t) \sigma(t) x(t)^{\gamma} dz(t)[$]

all of the explicit time functions will be defined by their Taylor series and the user will have to provide a general series formula in terms of coefficients and power of nth term of the Taylor/power series(In other words, you will have to define a general nth term of the series expansion of each time function) and you will also be able to control the order of the series expansion of the time functions meaning how many terms to include in each time function power/Taylor series for calculations of stochastic integral expansions. And then monte carlo simulation coefficients could be calculated to fourth order of expansion in terms of x-derivatives of drift and volatility terms and then converted to Ito-Taylor coefficients. If every thing goes well, I hope to complete the program in a week including checking it carefully and then post it here on my Wilmott thread.

Amin · March 22nd, 2020, 3:05 pm

The integral in fourth term is not straightforward at all but we can still do it. Here is the integral
[$] \int_0^t \int_0^{\tau} \exp(-\kappa s) dz(s) d \tau[$]
To solve this integral, we will have to first convert the exponential into its Taylor series and then integrate each Taylor term one by one.
This can be written as
[$] \int_0^t \int_0^{\tau} dz(s) d \tau-\kappa \int_0^t \int_0^{\tau} s dz(s) d \tau+.5 {\kappa}^2 \int_0^t \int_0^{\tau} s^2 dz(s) d \tau[$]
[$]-1/6 {\kappa}^3 \int_0^t \int_0^{\tau} s^3 dz(s) d \tau +1/24 {\kappa}^4 \int_0^t \int_0^{\tau} s^4 dz(s) d \tau[$]
each of the above integrals can be solved by our hermite iterated integral update formulas given below

[$]\int_0^t s^m H_{n}(z(s)) dz(s)= \frac {t^m H_{n+1}(z(t))}{ \sqrt{(n+1)} \sqrt{(2m+n+1)}} [$]
[$]\int_0^t s^m H_{n}(z(s)) ds = \frac {t^{m+1} H_{n}(z(t))}{(m+1)} (1 - \frac {\sqrt{n}}{\sqrt{(2m+n+2)}}) [$]

Sorry friends, the above single term update formula cannot be applied to the above series expansion where we have a collection of terms. This is because the update formula depends upon calculation of variance and the variance when calculated term by term does not match with correct variance calculated by squaring the whole expression. I was not careful when I wrote the formula. But I have calculated nice recursions for this formula when applied to collection of a term as in a series. My work towards the explicit time dependent SDE simulation project is going well and I hope to present the worked out programs in a few days. Again I have taken path of power series/Taylor series for the time functions in the SDE. Initial requirement is that these time functions should have a power series where different terms have integer powers on time. After the experience with this work, it might later be possible to have even non-integer powers but as a start, I am sticking with it. I will also allow users to choose relative time or absolute time for time functions. If nth monte carlo time step starts at time [$]t_n[$] relative time means [$]t-t_n[$](that resets at the start of every time interval) as opposed to absolute time which would be [$]t[$] as running time over all intervals.

Amin · March 23rd, 2020, 4:26 pm

This is for friends who want to solve the mean reverting stochastic volatility type equations with Ito-hermite method of moving and expanding/contracting grid. When we have the equation given as

[$]dx(t)=\kappa (\theta - x(t)) dt + \sigma x^{\gamma} dz(t) [$]

We used a transformed version of the above equation and simulated the dynamics of [$]w(t)=\frac{x(t)^{1-\gamma}}{(1-\gamma)}[$]. In original coordinates the dynamics of evolution of w(t) are given as

[$]dw(t)=\kappa \theta x^{-\gamma} dt - \kappa x^{1-\gamma} dt -.5 \gamma {\sigma}^2 x^{(\gamma-1)} dt + \sigma dz(t)[$]

Just like for mean reverting equations in original coordinates, we can try a suitable choice of exponential and decrease the number of drift terms from three to only two. And we know that our Ito-hermite method works remarkably well when there are only two terms in the drift. So here is the new process we simulate

[$]d[\exp(\kappa(1-\gamma) t) w(t)]=d[\exp(\kappa(1-\gamma) t) \frac{x(t)^{1-\gamma}}{(1-\gamma)}][$]
[$]=\exp(\kappa(1-\gamma) t) [\kappa \theta x^{-\gamma} dt - \kappa x^{1-\gamma} dt -.5 \gamma {\sigma}^2 x^{(\gamma-1)} dt + \sigma dz(t)][$]
[$]+\exp(\kappa(1-\gamma) t) \kappa x^{1-\gamma} dt[$]
once we cancel the identical terms with opposite signs, we get the equation
[$]d[\exp(\kappa(1-\gamma) t) w(t)][$]
[$]=\exp(\kappa(1-\gamma) t) [\kappa \theta x^{-\gamma} dt -.5 \gamma {\sigma}^2 x^{(\gamma-1)} dt + \sigma dz(t)][$]
solving the above equation in proper integral form, we get
[$]w(t)=\exp(-\kappa(1-\gamma) t) w(0)[$]
[$]+\exp(-\kappa(1-\gamma) t) \int_0^t \exp(\kappa(1-\gamma) s) [\kappa \theta x^{-\gamma} -.5 \gamma {\sigma}^2 x^{(\gamma-1)}] ds[$]
[$]+\exp(-\kappa(1-\gamma) t) \int_0^t \exp(\kappa(1-\gamma) s) \sigma dz(s)[$]

So we have decreased the number of drift terms from three to two in exchange for time dependence in the SDE. If we could solve for time exactly, I hope, we will be able to get very exact solution of the above type of mean reverting SDEs.
As I am trying to complete the program for monte carlo simulation of explicit time dependent SDEs, once I post that program in a few days, I will come back to integrating the above solution dynamics in our Ito-hermite framework for the solution of above mean reverting SDEs.

Breakthrough in the theory of stochastic differential equations and their simulation

Re: Breakthrough in the theory of stochastic differential equations and their simulation

Re: Breakthrough in the theory of stochastic differential equations and their simulation

Re: Breakthrough in the theory of stochastic differential equations and their simulation

Re: Breakthrough in the theory of stochastic differential equations and their simulation

Re: Breakthrough in the theory of stochastic differential equations and their simulation

Re: Breakthrough in the theory of stochastic differential equations and their simulation

Re: Breakthrough in the theory of stochastic differential equations and their simulation

Re: Breakthrough in the theory of stochastic differential equations and their simulation

Re: Breakthrough in the theory of stochastic differential equations and their simulation

Re: Breakthrough in the theory of stochastic differential equations and their simulation

Re: Breakthrough in the theory of stochastic differential equations and their simulation

Re: Breakthrough in the theory of stochastic differential equations and their simulation

Re: Breakthrough in the theory of stochastic differential equations and their simulation

Re: Breakthrough in the theory of stochastic differential equations and their simulation

Re: Breakthrough in the theory of stochastic differential equations and their simulation