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Amin
Topic Author
Posts: 2477
Joined: July 14th, 2002, 3:00 am

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

Please read the last two posts on previous page since they have far more significance than this post.
Sorry, I forgot to post this dependency function for the program in previous post.
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
Topic Author
Posts: 2477
Joined: July 14th, 2002, 3:00 am

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

I want to make this addition to my past few posts in recent days about appropriate calculations of barebone(first derivatives solution) solution to fokker-planck equation. What I wrote in previous few posts was the right calculation of first derivatives using adjacent "backward finite difference derivatives". We can do a very similar calculation with adjacent "forward finite difference derivatives" and then average both adjacent forward and adjacent backward calculations inside the squared expressions on both sides of the equation. This should possibly make the solution more accurate. I have yet not done this in matlab and hopefully try it today and then post a program with these calculations tomorrow. I hope it will add to accuracy.

Amin
Topic Author
Posts: 2477
Joined: July 14th, 2002, 3:00 am

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

Sorry for the delay in my post. I am not posting a new program but I would describe how I solved the equation using both forward and backward differences. It is very simple. I just solved the whole barebone equation with forward differences and then solved the whole barebone equation with backward differences and then averaged the two values to update w.  Some of my equations were occasionally being unstable at the boundary and therefore, many times, I would just replace forward and backward derivatives with central derivatives.
I am working on the program and hope to post a new program in a few days.

Amin
Topic Author
Posts: 2477
Joined: July 14th, 2002, 3:00 am

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

I am quickly posting this code which is based on derivatives derived from the analytic solution of Fokker-Planck equation. I works reasonably well only and accuracy deteriorates as value of kappa increases to larger values but this is only intermediate code before the final version. I will get an antipsychotic injection tomorrow and do not know if I will be able to work for many days after that.
Edit: I have just been injected a minute ago. Doctor sent a special guy from his psychiatric facility who brought his own injection. I already had talked to my mother that I will get an injection tomorrow and she agreed. But in that case I would buy an injection from market and a special injection with mind control chemicals could not be given to me. Doctor sent a special guy who brought his own injection so that an injection with MC chemicals could be given and it was impossible for me to refuse after the guy was already in our home with the injection. Please protest against these obvious wrongdoings by evil people in American defense and their cronies in Pakistan Army who receive tens of millions of dollars in bribes.
function [TrueMean,ItoHermiteMean,y_w,fy_w,wnStart] = FPERevisited044A()

%Copyright Ahsan Amin. Infiniti derivatives Technologies.
%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.
Tt=128;%16*16;     % Number of simulation levels. Terminal time= Tt*dt; //.125/32*32*16=2 year;
T=Tt*dt;
OrderA=4;  %
OrderM=4;  %
%dtM=.125/8;%Monte carlo time interval size dtM.
%TtM=8*8;%Monte carlo number of simulation intervals.
dtM=dt*1;
TtM=Tt/1;

dNn=.2/1;   % Normal density subdivisions width. would change with number of subdivisions
Nn=50;  % 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=.50;   % starting value of SDE
beta1=0.0;
beta2=1.0;   % Second drift term power.
gamma=.75;%50;   % volatility power.
kappa=.850;%.950;   %mean reversion parameter.
theta=.10;%mean reversion target
sigma0=.850;%Volatility value

%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.
%mu1=0;
%mu2=0;

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,d3wMu0dtdw3,c1,c2,c3,c4] = CalculateDriftAndVolOrder4(w,wnStart,Nn,YqCoeff0,Fp1,gamma,dt);

wPrev=w;
dw(wnStart:Nn)=c1(wnStart:Nn).*Z(wnStart:Nn) ;% ...
dw2(wnStart:Nn)=dw(wnStart:Nn).^2;
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,d2wdZ2A,d3wdZ3,d4wdZ4] = First4Derivatives2ndOrderEqSpaced(wnStart,Nn,dNn,w,Z);
dZdw(wnStart:Nn)=1.0./dwdZ(wnStart:Nn);

H2(wnStart:Nn)=(Z(wnStart:Nn).^2-1);

CTerm(wnStart:Nn)=(.25*d2wMu0dtdw2(wnStart:Nn).*dwdZ(wnStart:Nn).^2.*Z(wnStart:Nn)/4.0 - ...
1*dwMu0dtdw(wnStart:Nn).*d2wdZ2(wnStart:Nn));
Correction1(wnStart:Nn)=-.925/(22/7)^2.*CTerm(wnStart:Nn).*H2(wnStart:Nn);
Correction0(wnStart:Nn)=-.925/(22/7)^2.*(d2wdZ2(wnStart:Nn)-.25*dwdZ(wnStart:Nn).^2.*Z(wnStart:Nn)/4.0).*H2(wnStart:Nn);

%CTerm(wnStart:Nn)=wMu0dt(wnStart:Nn).*(dwdZ(wnStart:Nn).*(Z(wnStart:Nn))/4.0 - d2wdZ2(wnStart:Nn));
%Correction1(wnStart:Nn)=.5/(22/7)^2.*CTerm(wnStart:Nn).*H2(wnStart:Nn);
%Correction0(wnStart:Nn)=-2.0/(22/7)^2.*(d2wdZ2(wnStart:Nn)-1*dwdZ(wnStart:Nn).*(Z(wnStart:Nn))/4.0).*H2(wnStart:Nn);

CTerm(wnStart:Nn)=wMu0dt(wnStart:Nn).*(dwdZ(wnStart:Nn).*(Z(wnStart:Nn))/4.0 - d2wdZ2(wnStart:Nn));
Correction1(wnStart:Nn)=.5/(22/7)^2.*CTerm(wnStart:Nn).*H2(wnStart:Nn);
Correction0(wnStart:Nn)=-2.0/(22/7)^2.*(d2wdZ2(wnStart:Nn)-1*dwdZ(wnStart:Nn).*(Z(wnStart:Nn))/4.0).*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)

%u0(wnStart:Nn)=w(wnStart:Nn)-Z(wnStart:Nn).*dwdZ(wnStart:Nn);
%  signT1(wnStart:Nn)=sign(-3*Z(wnStart:Nn).^1.*dZdw(wnStart:Nn).*d2wdZ2(wnStart:Nn)*sigma0^2.*dt);%%.* ...
%sqrt(abs(3*Z(wnStart:Nn).^1.*dZdw(wnStart:Nn).*d2wdZ2(wnStart:Nn)*sigma0^2.*dt*sigma0^2*dt^2));
%  T1(wnStart:Nn)=(abs(3*Z(wnStart:Nn).^1.*dZdw(wnStart:Nn).*d2wdZ2(wnStart:Nn)*sigma0^2.*dt));

signT1(wnStart:Nn)=sign(-3*Z(wnStart:Nn).^1.*dZdw(wnStart:Nn).*d2wdZ2(wnStart:Nn)*sigma0^2.*dt);%%.* ...
%sqrt(abs(3*Z(wnStart:Nn).^1.*dZdw(wnStart:Nn).*d2wdZ2(wnStart:Nn)*sigma0^2.*dt*sigma0^2*dt^2));
T1(wnStart:Nn)=sqrt(abs(3*Z(wnStart:Nn).^1.*dZdw(wnStart:Nn).*d2wdZ2(wnStart:Nn)*sigma0^2.*dt));

signT2(wnStart:Nn)=sign(dwMu0dtdw(wnStart:Nn).*dwdZ(wnStart:Nn).^2);%%.* ...
%sqrt(abs(3*Z(wnStart:Nn).^1.*dZdw(wnStart:Nn).*d2wdZ2(wnStart:Nn)*sigma0^2.*dt*sigma0^2*dt^2));
T2(wnStart:Nn)=sqrt(abs(dwMu0dtdw(wnStart:Nn).*dwdZ(wnStart:Nn).^2));

dw(wnStart:Nn)=sigma0*sqrt(dt).*Z(wnStart:Nn);
dw2(wnStart:Nn)=sigma0^2*dt*Z(wnStart:Nn).^2;

%u0(wnStart:Nn)=w(wnStart:Nn)-Z(wnStart:Nn).*dwdZ(wnStart:Nn);
%signT1(wnStart:Nn)=sign(-3*Z(wnStart:Nn).^1.*dZdw(wnStart:Nn).*d2wdZ2(wnStart:Nn)*sigma0^2.*dt);%%.* ...
%sqrt(abs(3*Z(wnStart:Nn).^1.*dZdw(wnStart:Nn).*d2wdZ2(wnStart:Nn)*sigma0^2.*dt*sigma0^2*dt^2));
%T1(wnStart:Nn)=sqrt(abs(3*Z(wnStart:Nn).^1.*dZdw(wnStart:Nn).*d2wdZ2(wnStart:Nn)*sigma0^2.*dt));

%w_Z0=.5*w(NnMidh)+.5*w(NnMidl);
%[y0] = InterpolateOrderN6(N,x0,x1,x2,x3,x4,x5,x6,y1,y2,y3,y4,y5,y6)
[wMedian] = InterpolateOrderN6(6,0,Z(NnMidh),Z(NnMidh+1),Z(NnMidh+2),Z(NnMidl),Z(NnMidl-1),Z(NnMidl-2),w(NnMidh),w(NnMidh+1),w(NnMidh+2),w(NnMidl),w(NnMidl-1),w(NnMidl-2));

% w_Z0 is the value of w associated with Z=0 or median of the density.
% w_Z0 is not associated with any grid point so it is interpolated from grid
% points on both sides as above. There is no grid point at median.

% We start finding adjacent points away from median on both sides of
% the median of SDE density.
% w(NnMidh) (h for high; sorry for bad names) is the first grid point just right of
% median and associated adjacent point is w(Z=0) i.e value of w associated
% with the median.
% w(NnMidl) ( l for low) is the first grid point left of median and associated
% adjacent point is again the value of w associated with median.
% For w(NnMidh+1), adjacent grid point is w(NnMidh)
% For w(NnMidl-1), adjacent grid point is w(NnMidl)
% and so on for both sides further away from median.

dwdZB(wnStart:Nn)=0;
dwdZF(wnStart:Nn)=0;

dwdZB(NnMidh)=(w(NnMidh)-wMedian)/(.5*dNn);
dwdZB(NnMidl)=(w(NnMidl)-wMedian)/(-.5*dNn);
dwdZB(NnMidl-1:-1:wnStart)=(w(NnMidl-1:-1:wnStart)-w(NnMidl:-1:wnStart+1))/(-dNn);
dwdZB(NnMidh+1:Nn)=(w(NnMidh+1:Nn)-w(NnMidh:Nn-1))/dNn;

%dwdZF(NnMidh)=(w(NnMidh)-wMedian)/(.5*dNn);
%dwdZF(NnMidl)=(w(NnMidl)-wMedian)/(-.5*dNn);
dwdZF(NnMidl:-1:wnStart+1)=(w(NnMidl-1:-1:wnStart)-w(NnMidl:-1:wnStart+1))/(-dNn);
dwdZF(NnMidh:Nn-1)=(w(NnMidh+1:Nn)-w(NnMidh:Nn-1))/dNn;

dwdZB=dwdZ;
dwdZF=dwdZ;

wMu0dt(wnStart:Nn)+sqrt(.5)*d2wdZ2(wnStart:Nn))+ ...
-sqrt(dt).^1*signT1(wnStart:Nn).*T1(wnStart:Nn)+ ...
sqrt(4*dt).^1*signT2(wnStart:Nn).*T2(wnStart:Nn)+ ...
-sqrt(.5)*d2wdZ2(wnStart:Nn)+dw(wnStart:Nn)).* ...
-sqrt(.5)*d2wdZ2(wnStart:Nn)).* ...
-sqrt(.5)*d2wdZ2(wnStart:Nn)).^2+ ...
sign(+dw(wnStart:Nn)).*dw2(wnStart:Nn)- ...
(dt).*signT1(wnStart:Nn).*T1(wnStart:Nn).^2 + ...
4.*(dt).*signT2(wnStart:Nn).*T2(wnStart:Nn).^2));%+ ...

wMu0dt(wnStart+1:Nn-1)+sqrt(.5)*d2wdZ2(wnStart+1:Nn-1))+ ...
-sqrt(dt).^1*signT1(wnStart+1:Nn-1).*T1(wnStart+1:Nn-1)+ ...
sqrt(4*dt).^1*signT2(wnStart+1:Nn-1).*T2(wnStart+1:Nn-1)+ ...
-sqrt(.5)*d2wdZ2(wnStart+1:Nn-1)+dw(wnStart+1:Nn-1)).* ...
-sqrt(.5)*d2wdZ2(wnStart+1:Nn-1)).* ...
-sqrt(.5)*d2wdZ2(wnStart+1:Nn-1)).^2 + ...
sign(+dw(wnStart+1:Nn-1)).*dw2(wnStart+1:Nn-1)- ...
(dt).^1*signT1(wnStart+1:Nn-1).*T1(wnStart+1:Nn-1).^2 + ...
4*(dt)*signT2(wnStart+1:Nn-1).*T2(wnStart+1:Nn-1).^2));%+ ...

w1(wnStart)=wB(wnStart);
w1(Nn)=wB(Nn);
w1(wnStart+1:Nn-1)=.5*wF(wnStart+1:Nn-1)+.5*wB(wnStart+1:Nn-1);

w=w1;

w(wnStart:Nn)=w(wnStart:Nn)+sqrt(2)*(.75-gamma).*(d2wdZ2(wnStart:Nn)-.25*dwdZ(wnStart:Nn).^2).*H2(wnStart:Nn)*sigma0^2*dt;

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
%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
%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
% 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=.66;
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;

%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;

%
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);

%         w1(1:Nn-2)=w(1:Nn-2);
%         w2(1:Nn-2)=w(2:Nn-1);
%         w3(1:Nn-2)=w(3:Nn);
%         w(w1(:)>w2(:))=w2(w1(:)>w2(:))-.25*(w3(w1(:)>w2(:))-w2(w1(:)>w2(:)));

%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

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

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(300*gamma^2*4*sigma0);%Decrease the number of bins if the graph is too
[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(sprintf('Ito-Hermite Method VS Monte Carlo Density, x0 = %.2f,theta=%.2f,kappa=%.2f,gamma=%.3f,sigma=%.2f,T=%.2f', x0,theta,kappa,gamma,sigma0,T));%,sprintf('theta= %f', theta), sprintf('kappa = %f', kappa),sprintf('sigma = %f', sigma0),sprintf('T = %f', T));

%title('text',sprintf('x0 = %f', x0),sprintf('theta= %f', theta), sprintf('kappa = %f', kappa),sprintf('sigma = %f', sigma0),sprintf('T = %f', T))
%xlabel(a)
%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 output of the hard coded parameters.
Original process average from monte carlo
MCMean =
0.270321416687029
Original process average from our simulation
ItoHermiteMean =
0.270962577442225
true Mean only applicble to standard SV mean reverting type models otherwise disregard
TrueMean =
0.270965972779491
IndexMax =
575
red line is density of SDE from Ito-Hermite method, green is monte carlo.ans =
0.270965972779491

Here is the graph output of the above code with hardcoded parameters.

Amin
Topic Author
Posts: 2477
Joined: July 14th, 2002, 3:00 am

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

Dear friends I made some notes about simulation of SDEs using Ito-hermite method. In order to simulate the SDEs with this method, first we have to calculate the complete differential of square of terms as they appear inside the normal exponential. I will start with a general SDE in Lamperti form. Here is the form of the SDE
$dw_s=\mu(w_s) ds + \sigma dZ(s)$   Equation(1)

Let us suppose$\zeta_w$ is the mean of density when we want to calculate the one time step ahead value of $w_t$ to calculate  the density.
First of all we want to calculate the differential and later we will go to fokker-planck

$\int_0^t d\Big[ {(w_s -\zeta_w - \int_0^s \mu(w_v) dv )}^2 \Big] = {\sigma}^2 Z^2 \Delta t$     Equation(2)

Let us suppose that we indicate the integral of drift by uppercase letter as
$M_t=\int_0^t \mu(w_s) ds$   Equation(3)

Expanding the left side in the equation(2), we get

$\int_0^t d\Big[ {({w_s}^2+ {\zeta_w}^2 + {M_s}^2 - 2 w_s \zeta_w -2 w_s M_s + 2 \zeta_w M_s )} \Big]$

applying the differential we get

$\int_0^t \Big[ 2w_s dw_s +{\sigma}^2 ds +2 M_s dM_s - 2 \zeta_w dw_s -2 d[w_s M_s] + 2 \zeta_w dM_s \Big]$
The first integral given as
$\int_0^t \Big[ 2w_s dw_s +{\sigma}^2 ds \Big]$ is straightforward. The only two integrals that are relatively difficult are calculated below

$\int_0^t M_s dM_s$
In order to calculate the above integral, we first calculate
$M_s=\int_0^s \mu(w_v) dv$
$dM_s= \mu(w_s) ds$
$M_s= \mu(w_0) \int_0^s dv + \frac{\partial \mu(w_0)}{\partial w} \mu(w_0) \int_0^s \int_0^v du dv + \frac{\partial \mu(w_0)}{\partial w} \sigma \int_0^s \int_0^v dz(u) dv + .5 \frac{\partial^2 \mu(w_0)}{\partial w^2} {\sigma}^2 \int_0^s \int_0^v du dv$
$=\mu(w_0) s + \frac{\partial \mu(w_0)}{\partial w} \mu(w_0) \frac{s^2}{2} + \frac{\partial \mu(w_0)}{\partial w} \sigma (1-\frac{1}{\sqrt{3}}) s z(s)$
$+ .5 \frac{\partial^2 \mu(w_0)}{\partial w^2} {\sigma}^2 \frac{s^2}{2}$

Now
$\int_0^t M_s dM_s= \int_0^t\Big[ \mu(w_0) s + \frac{\partial \mu(w_0)}{\partial w} \mu(w_0) \frac{s^2}{2} + \frac{\partial \mu(w_0)}{\partial w} \sigma (1-\frac{1}{\sqrt{3}}) s z(s)$
$+ .5 \frac{\partial^2 \mu(w_0)}{\partial w^2} {\sigma}^2 \frac{s^2}{2} \Big ] dMs$
$= \int_0^t\Big[ {\mu(w_0)}^2 s ds+ \frac{\partial \mu(w_0)}{\partial w} {\mu(w_0)}^2 \frac{s^2}{2} ds+ \frac{\partial \mu(w_0)}{\partial w} \mu(w_0) \sigma (1-\frac{1}{\sqrt{3}}) s z(s) ds$
$+ .5 \frac{\partial^2 \mu(w_0)}{\partial w^2} \mu(w_0) {\sigma}^2 \frac{s^2}{2} ds \Big ]$
$= {\mu(w_0)}^2 \frac{t^2}{2}+ \frac{\partial \mu(w_0)}{\partial w} {\mu(w_0)}^2 \frac{t^3}{6}+ \frac{\partial \mu(w_0)}{\partial w} \mu(w_0) \sigma (1-\frac{1}{\sqrt{3}}) \frac{1}{2} (1-\frac{1}{\sqrt{5}}) t^2 z(s)$
$+ .5 \frac{\partial^2 \mu(w_0)}{\partial w^2} \mu(w_0) {\sigma}^2 \frac{t^3}{6}$

The other difficult and more important integral is calculated as
$= \int_0^t d[w_s M_s]= \int_0^t M_s dw_s +w_s dM_s$
We substitute the value of $M_s$ from above equation No.  into the above equation as
$= \int_0^t \Big[ \big[ \mu(w_0) s + \frac{\partial \mu(w_0)}{\partial w} \mu(w_0) \frac{s^2}{2} + \frac{\partial \mu(w_0)}{\partial w} \sigma (1-\frac{1}{\sqrt{3}}) s z(s)$
$+ .5 \frac{\partial^2 \mu(w_0)}{\partial w^2} {\sigma}^2 \frac{s^2}{2} \big] \big[\mu(w_0) ds + \sigma dz(s) \big]$
$+w_0 \mu(w_0) ds + w_0 \frac{\partial \mu(w_0)}{\partial w} \mu(w_0) s ds + w_0 \frac{\partial \mu(w_0)}{\partial w} \sigma z(s) ds$
$+w_0 .5 \frac{\partial^2 \mu(w_0)}{\partial w^2} {\sigma}^2 s ds \Big]$
$= \Big[ \big[ {\mu(w_0)}^2 \frac {t^2}{2} + \frac{\partial \mu(w_0)}{\partial w} {\mu(w_0)}^2 \frac{t^3}{6}$
$+\frac{\partial \mu(w_0)}{\partial w} \sigma (1-\frac{1}{\sqrt{3}}) \frac{1}{2} (1-\frac{1}{\sqrt{5}}) t^2 z(t)$
$+ .5 \frac{\partial^2 \mu(w_0)}{\partial w^2} \mu(w_0) {\sigma}^2 \frac{t^3}{6} \big]$
$+ \big[ {\mu(w_0)} \sigma (1-\frac {1}{\sqrt{3}}) t z(t) + \frac{\partial \mu(w_0)}{\partial w} {\mu(w_0)} .5 t^2 z(t)/\sqrt{2}/\sqrt{6}$
$+\frac{\partial \mu(w_0)}{\partial w} \sigma (1-\frac{1}{\sqrt{3}}) \frac{t ({z(t)}^2-1)}{2\sqrt{2}}$
$+ .5 \frac{\partial^2 \mu(w_0)}{\partial w^2} \mu(w_0) {\sigma}^2 \frac{t^2 z(t)}{\sqrt{2}}{\sqrt{6}} \big]$
$+w_0 \mu(w_0) t + w_0 \frac{\partial \mu(w_0)}{\partial w} \mu(w_0) \frac{t^2}{2} + w_0 \frac{\partial \mu(w_0)}{\partial w} \sigma (1-\frac{1}{\sqrt{3}}) t z(t)$
$+w_0 .5 \frac{\partial^2 \mu(w_0)}{\partial w^2} {\sigma}^2 \frac{t^2}{2} \Big]$

Just like I did in the copied post, I believe that we have to complete the square with our "adjacent terms" version of the equations. I will do some basic analysis for that. Please note that I have changed the notation somewhat from the copied post.

$\int_0^t d\Big[ {(w_n(s) -w_{n-1}(s) +Z_{n-1}\frac{dw}{dZ}- \int_0^s \mu(w(v)) dv )}^2 \Big] = {\sigma}^2 Z^2 \Delta t$     Equation(2)

I want to use the notation $\Delta w = w_n(s) -w_{n-1}(s)$
and therefore $\frac{d[\Delta w]}{dZ} = \frac{dw_n(s)}{dZ} -\frac{dw_{n-1}(s)}{dZ}$
and we can do change of variable analysis on  $\Delta w$ as
$d[\Delta w] = \frac{d[\Delta w]}{dZ} \frac{dZ}{dw} dw(s)$
$+ .5 (\frac{d[\Delta^2 w]}{dZ^2} {(\frac{dZ}{dw})}^2+ \frac{d[\Delta w]}{dZ} \frac{d^2 Z}{dw^2}) [dw(s)]^2$

using the above we can easily do Ito Change of variable for product terms in the square like
$d[{\Delta w} Z_{n-1}\frac{dw}{dZ}]$
and
$d[{\Delta w} \int_0^s \mu(w(v)) dv ]$
and
$d[Z_{n-1}\frac{dw}{dZ} \int_0^s \mu(w(v)) dv]$

Amin
Topic Author
Posts: 2477
Joined: July 14th, 2002, 3:00 am

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

I have not been able to make much progress on solution to FP equation. It is difficult to work when I am under effect of injections with mind control chemicals in them. I have decided to take a break from work on FPE and decided to complete my work on path integrals on transition probabilities grid. I have some very good ideas and I hope that I would be able to calculate arithmetic path integrals, dz-integrals and dt-integrals for general SDEs on transition probabilities grid. I really hope that this should be straightforward now with new ideas I have. There have been some more understanding about how mean of general SDEs should evolve and we could also incorporate that in the transition probabilities simulation framework for stochastic differential equations. I hope to start posting new programs in a few days.

Amin
Topic Author
Posts: 2477
Joined: July 14th, 2002, 3:00 am

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

I have been continuing work on both solution of FPE project and path-integrals on transition probabilities grid project. Here is some success on FPE project: I was able to take two sided derivatives in a single evolution equation for SDEs in their FPE solution. I was also able to take both first and second numerical derivatives in the calculations. Second numerical derivative was difficult since it was oscillatory due to interpolation issues that made the whole calculations unstable but for a large number of cases I have been able to make it work. And now for a huge number of cases out to several years and with a large range of mean reversion parameter, I have been able to get very precise solution for the densities of the mean reverting SDEs. I will be posting the code for FPE solution on this forum tomorrow.

Amin
Topic Author
Posts: 2477
Joined: July 14th, 2002, 3:00 am

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

Here is the new program for solution to Fokker-planck equation for SDEs.
function [TrueMean,ItoHermiteMean,y_w,fy_w,wnStart] = FPERevisited04Wilmott()

%Copyright Ahsan Amin. Infiniti derivatives Technologies.
%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.
Tt=128*8/4;%16*16;     % Number of simulation levels. Terminal time= Tt*dt; //.125/32*32*16=2 year;
T=Tt*dt;
OrderA=4;  %
OrderM=4;  %
%dtM=.125/8;%Monte carlo time interval size dtM.
%TtM=8*8;%Monte carlo number of simulation intervals.
dtM=dt*4;
TtM=Tt/4;

dNn=.2/1;   % Normal density subdivisions width. would change with number of subdivisions
Nn=50;  % 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=1.0;   % starting value of SDE
beta1=0.0;
beta2=1.0;   % Second drift term power.
gamma=.95;%50;   % volatility power.
kappa=1.250;%.950;   %mean reversion parameter.
theta=1.0;%mean reversion target
sigma0=.80;%Volatility value

%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.
%mu1=0;
%mu2=0;

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;
d2wdZ2A(1:Nn)=0;
d2wdZ2B(1:Nn)=0;
d2wdZ2C(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,d3wMu0dtdw3,c1,c2,c3,c4] = CalculateDriftAndVolOrder4(w,wnStart,Nn,YqCoeff0,Fp1,gamma,dt);

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

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

d2wdZ2(wnStart:Nn)=d2wdZ2(wnStart:Nn)+ d2wMu0dtdw2(wnStart:Nn).*dwdZ(wnStart:Nn).^2+dwMu0dtdw(wnStart:Nn).*d2wdZ2(wnStart:Nn)+.33*(Z(wnStart:Nn)).*sigma0^2*dt^2;;%+ ...

[dwdZ,d2wdZ2A] = First2Derivatives2ndOrderEqSpaced(wnStart,Nn,dNn,w,Z);

if (tt>64)
d2wdZ2=d2wdZ2A;
d2wdZ2(21:29)=d2wdZ2(20)*(30-(21:29))/10+d2wdZ2(30)*((21:29)-20)/10;
end
%If the calculation of the program blows, it means that above calculated
%derivative is unstable or bad. Plese replace with your favourite method of
%calculating this second derivative.

%plot((wnStart:Nn),d2wdZ2,'r',(wnStart:Nn),d2wdZ2A,'g');

%str=input('Look at comparison of d2wdZ2');

dZdw(wnStart:Nn)=1.0./dwdZ(wnStart:Nn);
H2(wnStart:Nn)=(Z(wnStart:Nn).^2-1);

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)

dw(wnStart:Nn)=sigma0*sqrt(dt).*Z(wnStart:Nn);
dw2(wnStart:Nn)=sigma0^2*dt*Z(wnStart:Nn).^2;

[wS] = InterpolateOrderN6(6,Z(1)-dNn,Z(1),Z(2),Z(3),Z(4),Z(5),Z(6),w(1),w(2),w(3),w(4),w(5),w(6));
[wE] = InterpolateOrderN6(6,Z(Nn)+dNn,Z(Nn),Z(Nn-1),Z(Nn-2),Z(Nn-3),Z(Nn-4),Z(Nn-5),w(Nn),w(Nn-1),w(Nn-2),w(Nn-3),w(Nn-4),w(Nn-5));

%wS in point interpolated to the left of the grid.
%wE in point interpolated to the right of the grid.

dwdZB(wnStart:Nn)=0;
dwdZF(wnStart:Nn)=0;

C11=.707;
C22=8*dt;%
C33=8*dt;%

C11*d2wdZ2(wnStart:Nn))+ ...
-C11*d2wdZ2(wnStart:Nn)+dw(wnStart:Nn)).* ...
-C11*d2wdZ2(wnStart:Nn)).* ...
-C11*d2wdZ2(wnStart:Nn)).^2+ ...
sign(+dw(wnStart:Nn)).*dw2(wnStart:Nn))))+ ...
-C33.*dwMu0dtdw(wnStart:Nn).*dwdZ(wnStart:Nn).^2 + ...
+C22.*3*Z(wnStart:Nn).^1.*dZdw(wnStart:Nn).*d2wdZ2(wnStart:Nn)*sigma0^2.*dt ;

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
%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
%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
% 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=.866;
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;

%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;

%
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);

%         w1(1:Nn-2)=w(1:Nn-2);
%         w2(1:Nn-2)=w(2:Nn-1);
%         w3(1:Nn-2)=w(3:Nn);
%         w(w1(:)>w2(:))=w2(w1(:)>w2(:))-.25*(w3(w1(:)>w2(:))-w2(w1(:)>w2(:)));

%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

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

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(300*gamma^2*4*sigma0);%Decrease the number of bins if the graph is too
[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(sprintf('Ito-Hermite Method VS Monte Carlo Density, x0 = %.2f,theta=%.2f,kappa=%.2f,gamma=%.3f,sigma=%.2f,T=%.2f', x0,theta,kappa,gamma,sigma0,T));%,sprintf('theta= %f', theta), sprintf('kappa = %f', kappa),sprintf('sigma = %f', sigma0),sprintf('T = %f', T));

%title('text',sprintf('x0 = %f', x0),sprintf('theta= %f', theta), sprintf('kappa = %f', kappa),sprintf('sigma = %f', sigma0),sprintf('T = %f', T))
%xlabel(a)
%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

One new supporting function is here:

function [dwdZ,d2wdZ2] = First2Derivatives2ndOrderEqSpaced(wnStart,Nn,dNn,w,Z)

[wS] = InterpolateOrderN6(5,Z(wnStart)-dNn,Z(wnStart),Z(wnStart+1),Z(wnStart+2),Z(wnStart+3),Z(wnStart+4),Z(wnStart+5),w(wnStart),w(wnStart+1),w(wnStart+2),w(wnStart+3),w(wnStart+4),w(wnStart+5));
[wE] = InterpolateOrderN6(5,Z(Nn)+dNn,Z(Nn),Z(Nn-1),Z(Nn-2),Z(Nn-3),Z(Nn-4),Z(Nn-5),w(Nn),w(Nn-1),w(Nn-2),w(Nn-3),w(Nn-4),w(Nn-5));

d2wdZ2(wnStart)=(wS-2*w(wnStart)+w(wnStart+1))/(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)=(wE-2*w(Nn)+w(Nn-1))/(dNn.^2);

dwdZ(wnStart)=(-1*wS+1*w(wnStart+1))/(2*dNn);
dwdZ(wnStart+1:Nn-1)=(-1*w(wnStart:Nn-2)+1*w(wnStart+2:Nn))/(2*dNn);
dwdZ(Nn)=(wE-w(Nn-1))/(2*dNn);

end


Here is the output of the program:
Original process average from monte carlo
MCMean =
1.001882559716443
Original process average from our simulation
ItoHermiteMean =
0.999990867727656
true Mean only applicble to standard SV mean reverting type models otherwise disregard
TrueMean =
1
IndexMax =
867

Here is the output graph after rescaling.

Amin
Topic Author
Posts: 2477
Joined: July 14th, 2002, 3:00 am

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

if (tt>64)
d2wdZ2=d2wdZ2A;
d2wdZ2(21:29)=d2wdZ2(20)*(30-(21:29))/10+d2wdZ2(30)*((21:29)-20)/10;
end

In my program, I used the above line of code which means that after 64 time intervals, replace my approximate analytical second derivative with interpolated numerical second derivative (d2wdZ2A, middle nine values are interpolated where we have more instability effect). I waited till 64 time intervals before replacing the second derivative so that numerical derivative becomes stable enough and the whole equation does not blow. This wait for 64 time intervals is very arbitrary and you could replace after 32 time intervals(or some other interval numbers value) and see if the solution remains stable(and you will see great improvement in solution for high mean reversions if solution remains stable). If exact second derivative is used everywhere, the solution will be far better.
Especially if you simulate the above solution for half year with high mean reversion and get bad results, try switching to numerical derivative after tt=32 or earlier if possible.

Amin
Topic Author
Posts: 2477
Joined: July 14th, 2002, 3:00 am

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

In the following program, I have made only three changes.

1) I have replaced in the following
if (tt>64)
d2wdZ2=d2wdZ2A;
d2wdZ2(21:29)=d2wdZ2(20)*(30-(21:29))/10+d2wdZ2(30)*((21:29)-20)/10;
end

with

f (tt>32)
d2wdZ2=d2wdZ2A;
d2wdZ2(21:29)=d2wdZ2(20)*(30-(21:29))/10+d2wdZ2(30)*((21:29)-20)/10;
end

2) and replaced following line
C33=8*dt;
with
C33=8*1.414*dt;

3) and replaced interpolation parameter
CTheta=.66;
with
CTheta=0;

I notice that density is exact as long as we follow exact numerical second derivative. Towards the right of the peak, the density is slightly inexact and that is because I am linearly interpolating numerical second derivative in that area. IF we can faithfully reproduce the second derivative, the solution of the fokker-planck equation will be exact with this method now even though we have not included the effect of third derivative at all yet. If the following program blows, it means that numerical second derivative has become unstable and my method to make numerical second derivative stable has failed. If you can make sure that your numerical second derivative is stable, you can get the program to run perfectly. But there will usually very slight and very negligible inaccuracy effect from not including third derivative.
Important thing is that all derivatives are from analytical derivatives of Fokker-planck equation that we obtained when we did a change of SDE density with normal density in an earlier post and I have not added any single fudge term at all in this solution.
Here I am posting the code with above changes.
function [TrueMean,ItoHermiteMean,y_w,fy_w,wnStart] = FPERevisited04Wilmott02()

%Copyright Ahsan Amin. Infiniti derivatives Technologies.
%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.
Tt=128*12/4;%16*16;     % Number of simulation levels. Terminal time= Tt*dt; //.125/32*32*16=2 year;
T=Tt*dt;
OrderA=4;  %
OrderM=4;  %
%dtM=.125/8;%Monte carlo time interval size dtM.
%TtM=8*8;%Monte carlo number of simulation intervals.
dtM=dt*4;
TtM=Tt/4;

dNn=.2/1;   % Normal density subdivisions width. would change with number of subdivisions
Nn=50;  % 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=1.0;   % starting value of SDE
beta1=0.0;
beta2=1.0;   % Second drift term power.
gamma=.55;%50;   % volatility power.
kappa=2.50;%.950;   %mean reversion parameter.
theta=1.0;%mean reversion target
sigma0=.80;%Volatility value

%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.
%mu1=0;
%mu2=0;

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;
d2wdZ2A(1:Nn)=0;
d2wdZ2B(1:Nn)=0;
d2wdZ2C(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,d3wMu0dtdw3,c1,c2,c3,c4] = CalculateDriftAndVolOrder4(w,wnStart,Nn,YqCoeff0,Fp1,gamma,dt);

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

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

d2wdZ2(wnStart:Nn)=d2wdZ2(wnStart:Nn)+ d2wMu0dtdw2(wnStart:Nn).*dwdZ(wnStart:Nn).^2+dwMu0dtdw(wnStart:Nn).*d2wdZ2(wnStart:Nn)+.33*(Z(wnStart:Nn)).*sigma0^2*dt^2;;%+ ...

[dwdZ,d2wdZ2A] = First2Derivatives2ndOrderEqSpaced(wnStart,Nn,dNn,w,Z);

if (tt>32)
d2wdZ2=d2wdZ2A;
d2wdZ2(21:29)=d2wdZ2(20)*(30-(21:29))/10+d2wdZ2(30)*((21:29)-20)/10;
end
%If the calculation of the program blows, it means that above calculated
%derivative is unstable or bad. Plese replace with your favourite method of
%calculating this second derivative.

plot((wnStart:Nn),d2wdZ2,'r',(wnStart:Nn),d2wdZ2A,'g');
tt
str=input('Look at comparison of d2wdZ2. Interpolation starts after tt==32');

dZdw(wnStart:Nn)=1.0./dwdZ(wnStart:Nn);
H2(wnStart:Nn)=(Z(wnStart:Nn).^2-1);

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)

dw(wnStart:Nn)=sigma0*sqrt(dt).*Z(wnStart:Nn);
dw2(wnStart:Nn)=sigma0^2*dt*Z(wnStart:Nn).^2;

[wS] = InterpolateOrderN6(6,Z(1)-dNn,Z(1),Z(2),Z(3),Z(4),Z(5),Z(6),w(1),w(2),w(3),w(4),w(5),w(6));
[wE] = InterpolateOrderN6(6,Z(Nn)+dNn,Z(Nn),Z(Nn-1),Z(Nn-2),Z(Nn-3),Z(Nn-4),Z(Nn-5),w(Nn),w(Nn-1),w(Nn-2),w(Nn-3),w(Nn-4),w(Nn-5));

%wS in point interpolated to the left of the grid.
%wE in point interpolated to the right of the grid.

dwdZB(wnStart:Nn)=0;
dwdZF(wnStart:Nn)=0;

C11=.707;
C22=8*dt;%
C33=8*1.414*dt;%

C11*d2wdZ2(wnStart:Nn))+ ...
-C11*d2wdZ2(wnStart:Nn)+dw(wnStart:Nn)).* ...
-C11*d2wdZ2(wnStart:Nn)).* ...
-C11*d2wdZ2(wnStart:Nn)).^2+ ...
sign(+dw(wnStart:Nn)).*dw2(wnStart:Nn))))+ ...
-C33.*dwMu0dtdw(wnStart:Nn).*dwdZ(wnStart:Nn).^2 + ...
+C22.*3*Z(wnStart:Nn).^1.*dZdw(wnStart:Nn).*d2wdZ2(wnStart:Nn)*sigma0^2.*dt ;

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
%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
%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
% 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=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;

%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;

%
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);

%         w1(1:Nn-2)=w(1:Nn-2);
%         w2(1:Nn-2)=w(2:Nn-1);
%         w3(1:Nn-2)=w(3:Nn);
%         w(w1(:)>w2(:))=w2(w1(:)>w2(:))-.25*(w3(w1(:)>w2(:))-w2(w1(:)>w2(:)));

%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

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

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(300*gamma^2*4*sigma0);%Decrease the number of bins if the graph is too
[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(sprintf('Ito-Hermite Method VS Monte Carlo Density, x0 = %.2f,theta=%.2f,kappa=%.2f,gamma=%.3f,sigma=%.2f,T=%.2f', x0,theta,kappa,gamma,sigma0,T));%,sprintf('theta= %f', theta), sprintf('kappa = %f', kappa),sprintf('sigma = %f', sigma0),sprintf('T = %f', T));

%title('text',sprintf('x0 = %f', x0),sprintf('theta= %f', theta), sprintf('kappa = %f', kappa),sprintf('sigma = %f', sigma0),sprintf('T = %f', T))
%xlabel(a)
%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 last output with this method.

Original process average from monte carlo
MCMean =
0.998946438546209
Original process average from our simulation
ItoHermiteMean =
0.999998003732751
true Mean only applicble to standard SV mean reverting type models otherwise disregard
TrueMean =
1
IndexMax =
291
red line is density of SDE from Ito-Hermite method, green is monte carlo.

Amin
Topic Author
Posts: 2477
Joined: July 14th, 2002, 3:00 am

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

This is another graph from program in previous post.

Another density. Parameters are in graph title.

Again mostly If the program in previous post blows, it means that numerical second derivative has become unstable and my method to make numerical second derivative stable has failed.

Amin
Topic Author
Posts: 2477
Joined: July 14th, 2002, 3:00 am

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

Post in progress.

I have recast exactly the same analytics in a 'special explicit form'. By 'special explicit form', I mean that now the evolution equation is of the form

$w_n(t+1)=w_n(t)+G(w(t),t)$  while earlier evolution equation was of the form  $w_n(t+1)=H(w(t),t)$

$w_n(t)$ appears at the start of equation's RHS and $w_n(t)$ disappears from the square-root part in RHS.

It is exactly the same analytics and will give precisely the same result as in old equation(as in the last program posted) but I recast in a new (here called 'special explicit') form. I have posted the new program. I will be explaining how I recast the old equation in this new form in a few hours.
function [TrueMean,ItoHermiteMean,y_w,fy_w,wnStart] = FPERevisited04Wilmott04()

%Copyright Ahsan Amin. Infiniti derivatives Technologies.
%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.
Tt=128*8/4;%16*16;     % Number of simulation levels. Terminal time= Tt*dt; //.125/32*32*16=2 year;
T=Tt*dt;
OrderA=4;  %
OrderM=4;  %
%dtM=.125/8;%Monte carlo time interval size dtM.
%TtM=8*8;%Monte carlo number of simulation intervals.
dtM=dt*4;
TtM=Tt/4;

dNn=.2/1;   % Normal density subdivisions width. would change with number of subdivisions
Nn=50;  % 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=1.0;   % starting value of SDE
beta1=0.0;
beta2=1.0;   % Second drift term power.
gamma=.85;%50;   % volatility power.
kappa=1.0;%.950;   %mean reversion parameter.
theta=1.0;%mean reversion target
sigma0=.80;%Volatility value

%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.
%mu1=0;
%mu2=0;

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;
d2wdZ2A(1:Nn)=0;
d2wdZ2B(1:Nn)=0;
d2wdZ2C(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,d3wMu0dtdw3,c1,c2,c3,c4] = CalculateDriftAndVolOrder4(w,wnStart,Nn,YqCoeff0,Fp1,gamma,dt);

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

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

d2wdZ2(wnStart:Nn)=d2wdZ2(wnStart:Nn)+ d2wMu0dtdw2(wnStart:Nn).*dwdZ(wnStart:Nn).^2+dwMu0dtdw(wnStart:Nn).*d2wdZ2(wnStart:Nn)+.33*(Z(wnStart:Nn)).*sigma0^2*dt^2;;%+ ...

[dwdZ,d2wdZ2A] = First2Derivatives2ndOrderEqSpaced(wnStart,Nn,dNn,w,Z);

if (tt>24)
d2wdZ2=d2wdZ2A;
d2wdZ2(21:29)=d2wdZ2(20)*(30-(21:29))/10+d2wdZ2(30)*((21:29)-20)/10;
%d2wdZ2(22:29)=d2wdZ2(21)*(30-(22:29))/9+d2wdZ2(28)*((23:27)-22)/5;
end
%If the calculation of the program blows, it means that above calculated
%derivative is unstable or bad. Plese replace with your favourite method of
%calculating this second derivative.

%plot((wnStart:Nn),d2wdZ2,'r',(wnStart:Nn),d2wdZ2A,'g');
%tt
%str=input('Look at comparison of d2wdZ2. Interpolation starts after tt=32');

dZdw(wnStart:Nn)=1.0./dwdZ(wnStart:Nn);
H2(wnStart:Nn)=(Z(wnStart:Nn).^2-1);

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)

dw(wnStart:Nn)=sigma0*sqrt(dt).*Z(wnStart:Nn);
dw2(wnStart:Nn)=sigma0^2*dt*Z(wnStart:Nn).^2;

[wS] = InterpolateOrderN6(6,Z(1)-dNn,Z(1),Z(2),Z(3),Z(4),Z(5),Z(6),w(1),w(2),w(3),w(4),w(5),w(6));
[wE] = InterpolateOrderN6(6,Z(Nn)+dNn,Z(Nn),Z(Nn-1),Z(Nn-2),Z(Nn-3),Z(Nn-4),Z(Nn-5),w(Nn),w(Nn-1),w(Nn-2),w(Nn-3),w(Nn-4),w(Nn-5));

%wS in point interpolated to the left of the grid.
%wE in point interpolated to the right of the grid.

dwdZB(wnStart:Nn)=0;
dwdZF(wnStart:Nn)=0;

C11=.707;
C22=8*dt;%
C33=8*1.414*dt;%

%             C11*d2wdZ2(wnStart:Nn))+ ...
%             -C11*d2wdZ2(wnStart:Nn)+dw(wnStart:Nn)).* ...
%             -C11*d2wdZ2(wnStart:Nn)).* ...
%             -C11*d2wdZ2(wnStart:Nn)).^2+ ...
%             sign(+dw(wnStart:Nn)).*dw2(wnStart:Nn))))+ ...
%             -C33.*dwMu0dtdw(wnStart:Nn).*dwdZ(wnStart:Nn).^2 + ...
%             +C22.*3*Z(wnStart:Nn).^1.*dZdw(wnStart:Nn).*d2wdZ2(wnStart:Nn)*sigma0^2.*dt ;

w(wnStart:Nn)=w(wnStart:Nn)+wMu0dt(wnStart:Nn)+((Term0(wnStart:Nn)+C11*d2wdZ2(wnStart:Nn))+ ...
sign(-Term0(wnStart:Nn)-C11*d2wdZ2(wnStart:Nn)+dw(wnStart:Nn)).* ...
sqrt(abs(sign(-Term0(wnStart:Nn)-C11*d2wdZ2(wnStart:Nn)).* ...
(-Term0(wnStart:Nn)-C11*d2wdZ2(wnStart:Nn)).^2+ ...
sign(+dw(wnStart:Nn)).*dw2(wnStart:Nn))))+ ...
-C33.*dwMu0dtdw(wnStart:Nn).*dwdZ(wnStart:Nn).^2 + ...
+C22.*3*Z(wnStart:Nn).^1.*dZdw(wnStart:Nn).*d2wdZ2(wnStart:Nn)*sigma0^2.*dt;

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
%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
%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
% 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=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;

%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;

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);

%         w1(1:Nn-2)=w(1:Nn-2);
%         w2(1:Nn-2)=w(2:Nn-1);
%         w3(1:Nn-2)=w(3:Nn);
%         w(w1(:)>w2(:))=w2(w1(:)>w2(:))-.25*(w3(w1(:)>w2(:))-w2(w1(:)>w2(:)));

%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

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

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(300*gamma^2*4*sigma0);%Decrease the number of bins if the graph is too
[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(sprintf('Ito-Hermite Method VS Monte Carlo Density, x0 = %.2f,theta=%.2f,kappa=%.2f,gamma=%.3f,sigma=%.2f,T=%.2f', x0,theta,kappa,gamma,sigma0,T));%,sprintf('theta= %f', theta), sprintf('kappa = %f', kappa),sprintf('sigma = %f', sigma0),sprintf('T = %f', T));

%title('text',sprintf('x0 = %f', x0),sprintf('theta= %f', theta), sprintf('kappa = %f', kappa),sprintf('sigma = %f', sigma0),sprintf('T = %f', T))
%xlabel(a)
%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 last output of the program.
MCMean =
1.001678982032144
Original process average from our simulation
ItoHermiteMean =
0.999992600280595
true Mean only applicble to standard SV mean reverting type models otherwise disregard
TrueMean =
1
IndexMax =
695
And here is the output graph

Amin
Topic Author
Posts: 2477
Joined: July 14th, 2002, 3:00 am

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

I have made three minor changes to the matlab program which allow a larger range of parameters to be executed with the current program. I am copying new comments made on the program due to new changes here and then upload the new program.

%Change 1:7/25/2020: I have very heavily weighted the second derivative to
%the left of density peak with an exponential and it allows far larger
%range of parameters with stable execution without sacrificing much
%accuracy at all. Of course, it is just ad hoc.

I have made this weighting as in above comment from -2 to -5 normal SDs. After negative 2 till negative five  standard deviations.

%Change 2:7/25/2020:%I have added a last term from quadratic variation between drift
%and w terms. I am not hundred percent sure if we need it or not.

%Change 3:7/25/2020: I have improved zero correction in above.

Here is the new program which will work for many more parameter values than the earlier program. I have disabled mean-correction in this program code.
function [TrueMean,ItoHermiteMean,y_w,fy_w,wnStart] = FPERevisited044AAA02()

%Copyright Ahsan Amin. Infiniti derivatives Technologies.
%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.
Tt=128*8/4;%16*16;     % Number of simulation levels. Terminal time= Tt*dt; //.125/32*32*16=2 year;
T=Tt*dt;
OrderA=4;  %
OrderM=4;  %
%dtM=.125/8;%Monte carlo time interval size dtM.
%TtM=8*8;%Monte carlo number of simulation intervals.
dtM=dt*4;
TtM=Tt/4;

dNn=.2/1;   % Normal density subdivisions width. would change with number of subdivisions
Nn=50;  % 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=1.0;   % starting value of SDE
beta1=0.0;
beta2=1.0;   % Second drift term power.
gamma=.75;%50;   % volatility power.
kappa=2.0;%.950;   %mean reversion parameter.
theta=1.0;%mean reversion target
sigma0=.80;%Volatility value

%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.
%mu1=0;
%mu2=0;

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;
d2wdZ2A(1:Nn)=0;
d2wdZ2B(1:Nn)=0;
d2wdZ2C(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,d3wMu0dtdw3,c1,c2,c3,c4] = CalculateDriftAndVolOrder4(w,wnStart,Nn,YqCoeff0,Fp1,gamma,dt);

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

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

d2wdZ2(wnStart:Nn)=d2wdZ2(wnStart:Nn)+ d2wMu0dtdw2(wnStart:Nn).*dwdZ(wnStart:Nn).^2+dwMu0dtdw(wnStart:Nn).*d2wdZ2(wnStart:Nn)+.33*(Z(wnStart:Nn)).*sigma0^2*dt^2;;%+ ...

[dwdZ,d2wdZ2A] = First2Derivatives2ndOrderEqSpaced(wnStart,Nn,dNn,w,Z);

if (tt>24)
d2wdZ2=d2wdZ2A;
d2wdZ2(21:29)=d2wdZ2(20)*(30-(21:29))/10+d2wdZ2(30)*((21:29)-20)/10;

d2wdZ2(1:15)=d2wdZ2(1:15).*exp(-32*kappa^2*(1-gamma).*dNn*(15-(1:15)));
%Change 1:7/25/2020: I have very heavily weighted the second derivative to
%the left of density peak with an exponential and it allows far larger
%range of parameters with stable execution without sacrificing much
%accuracy. Of course, it is just ad hoc.
end
%If the calculation of the program blows, it means that above calculated
%derivative is unstable or bad. Plese replace with your favourite method of
%calculating this second derivative.

%plot((wnStart:Nn),d2wdZ2,'r',(wnStart:Nn),d2wdZ2A,'g');
%tt
%str=input('Look at comparison of d2wdZ2. Interpolation starts after tt=32');

dZdw(wnStart:Nn)=1.0./dwdZ(wnStart:Nn);
H2(wnStart:Nn)=(Z(wnStart:Nn).^2-1);

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)

dw(wnStart:Nn)=sigma0*sqrt(dt).*Z(wnStart:Nn);
dw2(wnStart:Nn)=sigma0^2*dt*Z(wnStart:Nn).^2;

[wS] = InterpolateOrderN6(6,Z(1)-dNn,Z(1),Z(2),Z(3),Z(4),Z(5),Z(6),w(1),w(2),w(3),w(4),w(5),w(6));
[wE] = InterpolateOrderN6(6,Z(Nn)+dNn,Z(Nn),Z(Nn-1),Z(Nn-2),Z(Nn-3),Z(Nn-4),Z(Nn-5),w(Nn),w(Nn-1),w(Nn-2),w(Nn-3),w(Nn-4),w(Nn-5));

%wS in point interpolated to the left of the grid.
%wE in point interpolated to the right of the grid.

dwdZB(wnStart:Nn)=0;
dwdZF(wnStart:Nn)=0;

C11(wnStart:Nn)=.707;%dwdZ(wnStart:Nn);%.707000;
C22=4*dt;%
C33=8*dt*sqrt(2);%
C44=1.0;
%  C22=.5*sigma0*sqrt(dt);%
%  C33=sigma0*sqrt(dt);%

%             C11*d2wdZ2(wnStart:Nn))+ ...
%             -C11*d2wdZ2(wnStart:Nn)+dw(wnStart:Nn)).* ...
%             -C11*d2wdZ2(wnStart:Nn)).* ...
%             -C11*d2wdZ2(wnStart:Nn)).^2+ ...
%             sign(+dw(wnStart:Nn)).*dw2(wnStart:Nn))))+ ...
%             -C33.*dwMu0dtdw(wnStart:Nn).*dwdZ(wnStart:Nn).^2 + ...
%             +C22.*3*Z(wnStart:Nn).^1.*dZdw(wnStart:Nn).*d2wdZ2(wnStart:Nn)*sigma0^2.*dt ;

w(wnStart:Nn)=w(wnStart:Nn)+wMu0dt(wnStart:Nn)+((Term0(wnStart:Nn)+C11(wnStart:Nn).*d2wdZ2(wnStart:Nn))+ ...
sign(-Term0(wnStart:Nn)-C11(wnStart:Nn).*d2wdZ2(wnStart:Nn)+dw(wnStart:Nn)).* ...
sqrt(abs(sign(-Term0(wnStart:Nn)-C11(wnStart:Nn).*d2wdZ2(wnStart:Nn)).* ...
(-Term0(wnStart:Nn)-C11(wnStart:Nn).*d2wdZ2(wnStart:Nn)).^2+ ...
sign(+dw(wnStart:Nn)).*dw2(wnStart:Nn))))+ ...
-C33.*dwMu0dtdw(wnStart:Nn).*dwdZ(wnStart:Nn).^2 + ...
+C22.*3*Z(wnStart:Nn).^1.*dZdw(wnStart:Nn).*d2wdZ2(wnStart:Nn)*sigma0^2.*dt + ...
+(1-1/sqrt(3))*dwMu0dtdw(wnStart:Nn).*sigma0^2*dt;

%Change 2:7/25/2020:%I have added a last term from quadratic variation between drift
%and w terms. I am not very sure if we need it or not.

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
%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
%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
% 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=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;

%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;

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);

[wE] = InterpolateOrderN6(6,Z(Nn)+dNn,Z(Nn),Z(Nn-1),Z(Nn-2),Z(Nn-3),Z(Nn-4),Z(Nn-5),w(Nn),w(Nn-1),w(Nn-2),w(Nn-3),w(Nn-4),w(Nn-5));

w1(1:Nn-1)=w(1:Nn-1);
w1(Nn)=w(Nn);
w2(1:Nn-1)=w(2:Nn);
w2(Nn)=wE;

w(w1(:)>w2(:))=0;%Be careful;might not universally hold;
%Change 3:7/25/2020: I have improved zero correction in above.
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

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

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(300*gamma^2*4*sigma0);%Decrease the number of bins if the graph is too
[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(sprintf('Ito-Hermite Method VS Monte Carlo Density, x0 = %.2f,theta=%.2f,kappa=%.2f,gamma=%.3f,sigma=%.2f,T=%.2f', x0,theta,kappa,gamma,sigma0,T));%,sprintf('theta= %f', theta), sprintf('kappa = %f', kappa),sprintf('sigma = %f', sigma0),sprintf('T = %f', T));

%title('text',sprintf('x0 = %f', x0),sprintf('theta= %f', theta), sprintf('kappa = %f', kappa),sprintf('sigma = %f', sigma0),sprintf('T = %f', T))
%xlabel(a)
%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 output of the program. Mean-correction is disabled.
MCMean =
1.000838760266297
Original process average from our simulation
ItoHermiteMean =
0.998849108605160
true Mean only applicble to standard SV mean reverting type models otherwise disregard
TrueMean =
1
IndexMax =
541

Here is the output graph of the program.

Amin
Topic Author
Posts: 2477
Joined: July 14th, 2002, 3:00 am

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

Amin
Topic Author
Posts: 2477
Joined: July 14th, 2002, 3:00 am

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

Post in progress.

%Change 7/27/2020
%I have switched to Lagrange interpolation by taking four points on both
%sides of instability and using 8th order interpolation of five points in
%mid between the eight points on sides. This works far better and you can
%now use nuemrical 2nd derivative without any modification for a large
%range of parameters. It truly improves the fit to true density when it
%works.

function [TrueMean,ItoHermiteMean,y_w,fy_w,wnStart] = FPERevisited044Wilmott08()

%Copyright Ahsan Amin. Infiniti derivatives Technologies.
%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.
Tt=128*8/4;%16*16;     % Number of simulation levels. Terminal time= Tt*dt; //.125/32*32*16=2 year;
T=Tt*dt;
OrderA=4;  %
OrderM=4;  %
%dtM=.125/8;%Monte carlo time interval size dtM.
%TtM=8*8;%Monte carlo number of simulation intervals.
dtM=dt*4;
TtM=Tt/4;

dNn=.2/1;   % Normal density subdivisions width. would change with number of subdivisions
Nn=50;  % 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=1.0;   % starting value of SDE
beta1=0.0;
beta2=1.0;   % Second drift term power.
gamma=.75;%50;   % volatility power.
kappa=2.0;%.950;   %mean reversion parameter.
theta=.250;%mean reversion target
sigma0=.70;%Volatility value

%you can specify any general mu1 and mu2 and beta1 and beta2.

mu1=+1*theta*kappa;   %first drift coefficient.
mu2=-1*kappa;    % Second drift coefficient.
%mu1=0;
%mu2=0;

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;
d2wdZ2A(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,d3wMu0dtdw3,c1,c2,c3,c4] = CalculateDriftAndVolOrder4(w,wnStart,Nn,YqCoeff0,Fp1,gamma,dt);

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

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

d2wdZ2(wnStart:Nn)=d2wdZ2(wnStart:Nn)+ d2wMu0dtdw2(wnStart:Nn).*dwdZ(wnStart:Nn).^2+dwMu0dtdw(wnStart:Nn).*d2wdZ2(wnStart:Nn)+.33*(Z(wnStart:Nn)).*sigma0^2*dt^2;;%+ ...

[dwdZ,d2wdZ2A] = First2Derivatives2ndOrderEqSpaced(wnStart,Nn,dNn,w,Z);

if (tt>16)
d2wdZ2=d2wdZ2A;

%d2wdZ2(21:29)=d2wdZ2(20)*(30-(21:29))/10+d2wdZ2(30)*((21:29)-20)/10;

d2wdZ2(1:15)=d2wdZ2(1:15).*exp(-32*kappa^2*(1-gamma).*dNn*(15-(1:15)));
%d2wdZ2(1:11)=d2wdZ2(1:11).*exp(-32*kappa^2*(1-gamma).*dNn*(11-(1:11)));
%Change 1:7/25/2020: I have very heavily weighted the second derivative to
%the left of density peak with an exponential and it allows far larger
%range of parameters with stable execution without sacrificing much
%accuracy. Of course, it is just ad hoc.
end
%If the calculation of the program blows, it means that above calculated
%derivative is unstable or bad. Plese replace with your favourite method of
%calculating this second derivative.

%plot((wnStart:Nn),d2wdZ2,'r',(wnStart:Nn),d2wdZ2A,'g');
%tt
%str=input('Look at comparison of d2wdZ2. Interpolation starts after tt=32');

dZdw(wnStart:Nn)=1.0./dwdZ(wnStart:Nn);
H2(wnStart:Nn)=(Z(wnStart:Nn).^2-1);

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)

dw(wnStart:Nn)=sigma0*sqrt(dt).*Z(wnStart:Nn);
dw2(wnStart:Nn)=sigma0^2*dt*Z(wnStart:Nn).^2;

[wS] = InterpolateOrderN6(6,Z(1)-dNn,Z(1),Z(2),Z(3),Z(4),Z(5),Z(6),w(1),w(2),w(3),w(4),w(5),w(6));
[wE] = InterpolateOrderN6(6,Z(Nn)+dNn,Z(Nn),Z(Nn-1),Z(Nn-2),Z(Nn-3),Z(Nn-4),Z(Nn-5),w(Nn),w(Nn-1),w(Nn-2),w(Nn-3),w(Nn-4),w(Nn-5));

%wS in point interpolated to the left of the grid.
%wE in point interpolated to the right of the grid.

dwdZB(wnStart:Nn)=0;
dwdZF(wnStart:Nn)=0;

C11(wnStart:Nn)=.707; %dwdZ(wnStart:Nn);%.707000;
C22=1*dt;%
C33=4*dt*sqrt(2*2*2);%
%  C22=.5*sigma0*sqrt(dt);%
%  C33=sigma0*sqrt(dt);%

w(wnStart:Nn)=w(wnStart:Nn)+wMu0dt(wnStart:Nn)+((Term0(wnStart:Nn)+C11(wnStart:Nn).*d2wdZ2(wnStart:Nn))+ ...
sign(-Term0(wnStart:Nn)-C11(wnStart:Nn).*d2wdZ2(wnStart:Nn)+dw(wnStart:Nn)).* ...
sqrt(abs(sign(-Term0(wnStart:Nn)-C11(wnStart:Nn).*d2wdZ2(wnStart:Nn)).* ...
(-Term0(wnStart:Nn)-C11(wnStart:Nn).*d2wdZ2(wnStart:Nn)).^2+ ...
sign(+dw(wnStart:Nn)).*dw2(wnStart:Nn))))+ ...
-C33.*dwMu0dtdw(wnStart:Nn).*dwdZ(wnStart:Nn).^2 + ...
+C22.*3*Z(wnStart:Nn).^1.*dZdw(wnStart:Nn).*d2wdZ2(wnStart:Nn)*sigma0^2.*dt + ...
+0*(1-1/sqrt(3))*dwMu0dtdw(wnStart:Nn).*sigma0^2*dt - ...
0*.125/2*.125*d2wdZ2(wnStart:Nn).*(w(wnStart:Nn)-Term0(wnStart:Nn))*dt;

%Change 2:7/25/2020:%I have added a last term from quadratic variation between drift
%and w terms. I am not very sure if we need it or not.

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
%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
%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
% 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.
%Change 7/27/2020
%I have switched to Lagrange interpolation by taking four points on both
%sides of instability and using 8th order interpolation of five points in
%mid between the eight points on sides. This works far better and you can
%now use nuemrical 2nd derivative without any modification for a large
%range of parameters. It truly improves the fit to true density when it
%works.

% Good interpolation below
w(NnMidh+2)=InterpolateOrderN8(8,Z(NnMidh+2),Z(NnMidh+3),Z(NnMidh+4),Z(NnMidh+5),Z(NnMidh+6),Z(NnMidl-2),Z(NnMidl-3),Z(NnMidl-4),Z(NnMidl-5),w(NnMidh+3),w(NnMidh+4),w(NnMidh+5),w(NnMidh+6),w(NnMidl-2),w(NnMidl-3),w(NnMidl-4),w(NnMidl-5));
w(NnMidh+1)=InterpolateOrderN8(8,Z(NnMidh+1),Z(NnMidh+3),Z(NnMidh+4),Z(NnMidh+5),Z(NnMidh+6),Z(NnMidl-2),Z(NnMidl-3),Z(NnMidl-4),Z(NnMidl-5),w(NnMidh+3),w(NnMidh+4),w(NnMidh+5),w(NnMidh+6),w(NnMidl-2),w(NnMidl-3),w(NnMidl-4),w(NnMidl-5));
w(NnMidh)=InterpolateOrderN8(8,Z(NnMidh),Z(NnMidh+3),Z(NnMidh+4),Z(NnMidh+5),Z(NnMidh+6),Z(NnMidl-2),Z(NnMidl-3),Z(NnMidl-4),Z(NnMidl-5),w(NnMidh+3),w(NnMidh+4),w(NnMidh+5),w(NnMidh+6),w(NnMidl-2),w(NnMidl-3),w(NnMidl-4),w(NnMidl-5));
w(NnMidl)=InterpolateOrderN8(8,Z(NnMidl),Z(NnMidh+3),Z(NnMidh+4),Z(NnMidh+5),Z(NnMidh+6),Z(NnMidl-2),Z(NnMidl-3),Z(NnMidl-4),Z(NnMidl-5),w(NnMidh+3),w(NnMidh+4),w(NnMidh+5),w(NnMidh+6),w(NnMidl-2),w(NnMidl-3),w(NnMidl-4),w(NnMidl-5));
w(NnMidl-1)=InterpolateOrderN8(8,Z(NnMidl-1),Z(NnMidh+3),Z(NnMidh+4),Z(NnMidh+5),Z(NnMidh+6),Z(NnMidl-2),Z(NnMidl-3),Z(NnMidl-4),Z(NnMidl-5),w(NnMidh+3),w(NnMidh+4),w(NnMidh+5),w(NnMidh+6),w(NnMidl-2),w(NnMidl-3),w(NnMidl-4),w(NnMidl-5));

%
%
%
%     CTheta=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;
%
% %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;
%
%
%
%
%
%         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);

[wE] = InterpolateOrderN6(6,Z(Nn)+dNn,Z(Nn),Z(Nn-1),Z(Nn-2),Z(Nn-3),Z(Nn-4),Z(Nn-5),w(Nn),w(Nn-1),w(Nn-2),w(Nn-3),w(Nn-4),w(Nn-5));

w1(1:Nn-1)=w(1:Nn-1);
w1(Nn)=w(Nn);
w2(1:Nn-1)=w(2:Nn);
w2(Nn)=wE;

w(w1(:)>w2(:))=0;%Be careful;might not universally hold;
%Change 3:7/25/2020: I have improved zero correction in above.
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

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

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(300*gamma^2*4*sigma0);%Decrease the number of bins if the graph is too
[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(sprintf('Ito-Hermite Method VS Monte Carlo Density, x0 = %.2f,theta=%.2f,kappa=%.2f,gamma=%.3f,sigma=%.2f,T=%.2f', x0,theta,kappa,gamma,sigma0,T));%,sprintf('theta= %f', theta), sprintf('kappa = %f', kappa),sprintf('sigma = %f', sigma0),sprintf('T = %f', T));

%title('text',sprintf('x0 = %f', x0),sprintf('theta= %f', theta), sprintf('kappa = %f', kappa),sprintf('sigma = %f', sigma0),sprintf('T = %f', T))
%xlabel(a)
%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

This is a new function you will need.
function [y0] = InterpolateOrderN8(N,x0,x1,x2,x3,x4,x5,x6,x7,x8,y1,y2,y3,y4,y5,y6,y7,y8)

if(N==8)
y0=(x0-x2)*(x0-x3)*(x0-x4)*(x0-x5)*(x0-x6)*(x0-x7)*(x0-x8)/((x1-x2).*(x1-x3).*(x1-x4).*(x1-x5).*(x1-x6).*(x1-x7).*(x1-x8))*y1+ ...
(x0-x1)*(x0-x3)*(x0-x4)*(x0-x5)*(x0-x6)*(x0-x7)*(x0-x8)/((x2-x1).*(x2-x3).*(x2-x4).*(x2-x5).*(x2-x6).*(x2-x7).*(x2-x8))*y2+ ...
(x0-x1)*(x0-x2)*(x0-x4)*(x0-x5)*(x0-x6)*(x0-x7)*(x0-x8)/((x3-x1).*(x3-x2).*(x3-x4).*(x3-x5).*(x3-x6).*(x3-x7).*(x3-x8))*y3+ ...
(x0-x1)*(x0-x2)*(x0-x3)*(x0-x5)*(x0-x6)*(x0-x7)*(x0-x8)/((x4-x1).*(x4-x2).*(x4-x3).*(x4-x5).*(x4-x6).*(x4-x7).*(x4-x8))*y4+ ...
(x0-x1)*(x0-x2)*(x0-x3)*(x0-x4)*(x0-x6)*(x0-x7)*(x0-x8)/((x5-x1).*(x5-x2).*(x5-x3).*(x5-x4).*(x5-x6).*(x5-x7).*(x5-x8))*y5+ ...
(x0-x1)*(x0-x2)*(x0-x3)*(x0-x4)*(x0-x5)*(x0-x7)*(x0-x8)/((x6-x1).*(x6-x2).*(x6-x3).*(x6-x4).*(x6-x5).*(x6-x7).*(x6-x8))*y6+ ...
(x0-x1)*(x0-x2)*(x0-x3)*(x0-x4)*(x0-x5)*(x0-x6)*(x0-x8)/((x7-x1).*(x7-x2).*(x7-x3).*(x7-x4).*(x7-x5).*(x7-x6).*(x7-x8))*y7+ ...
(x0-x1)*(x0-x2)*(x0-x3)*(x0-x4)*(x0-x5)*(x0-x6)*(x0-x7)/((x8-x1).*(x8-x2).*(x8-x3).*(x8-x4).*(x8-x5).*(x8-x6).*(x8-x7))*y8;

end

if(N==8)
y0=(x0-x2)*(x0-x3)*(x0-x4)*(x0-x5)*(x0-x6)*(x0-x7)*(x0-x8)/((x1-x2).*(x1-x3).*(x1-x4).*(x1-x5).*(x1-x6).*(x1-x7).*(x1-x8))*y1+ ...
(x0-x1)*(x0-x3)*(x0-x4)*(x0-x5)*(x0-x6)*(x0-x7)*(x0-x8)/((x2-x1).*(x2-x3).*(x2-x4).*(x2-x5).*(x2-x6).*(x2-x7).*(x2-x8))*y2+ ...
(x0-x1)*(x0-x2)*(x0-x4)*(x0-x5)*(x0-x6)*(x0-x7)*(x0-x8)/((x3-x1).*(x3-x2).*(x3-x4).*(x3-x5).*(x3-x6).*(x3-x7).*(x3-x8))*y3+ ...
(x0-x1)*(x0-x2)*(x0-x3)*(x0-x5)*(x0-x6)*(x0-x7)*(x0-x8)/((x4-x1).*(x4-x2).*(x4-x3).*(x4-x5).*(x4-x6).*(x4-x7).*(x4-x8))*y4+ ...
(x0-x1)*(x0-x2)*(x0-x3)*(x0-x4)*(x0-x6)*(x0-x7)*(x0-x8)/((x5-x1).*(x5-x2).*(x5-x3).*(x5-x4).*(x5-x6).*(x5-x7).*(x5-x8))*y5+ ...
(x0-x1)*(x0-x2)*(x0-x3)*(x0-x4)*(x0-x5)*(x0-x7)*(x0-x8)/((x6-x1).*(x6-x2).*(x6-x3).*(x6-x4).*(x6-x5).*(x6-x7).*(x6-x8))*y6+ ...
(x0-x1)*(x0-x2)*(x0-x3)*(x0-x4)*(x0-x5)*(x0-x6)*(x0-x8)/((x7-x1).*(x7-x2).*(x7-x3).*(x7-x4).*(x7-x5).*(x7-x6).*(x7-x8))*y7+ ...
(x0-x1)*(x0-x2)*(x0-x3)*(x0-x4)*(x0-x5)*(x0-x6)*(x0-x7)/((x8-x1).*(x8-x2).*(x8-x3).*(x8-x4).*(x8-x5).*(x8-x6).*(x8-x7))*y8;

end

if(N==7)
y0=(x0-x2)*(x0-x3)*(x0-x4)*(x0-x5)*(x0-x6)*(x0-x7)/((x1-x).*(x1-x3).*(x1-x4).*(x1-x5).*(x1-x6).*(x1-x7))*y1+ ...
(x0-x1)*(x0-x3)*(x0-x4)*(x0-x5)*(x0-x6)*(x0-x7)/((x2-x1).*(x2-x3).*(x2-x4).*(x2-x5).*(x2-x6).*(x2-x7))*y2+ ...
(x0-x1)*(x0-x2)*(x0-x4)*(x0-x5)*(x0-x6)*(x0-x7)/((x3-x1).*(x3-x2).*(x3-x4).*(x3-x5).*(x3-x6).*(x3-x7))*y3+ ...
(x0-x1)*(x0-x2)*(x0-x3)*(x0-x5)*(x0-x6)*(x0-x7)/((x4-x1).*(x4-x2).*(x4-x3).*(x4-x5).*(x4-x6).*(x4-x7))*y4+ ...
(x0-x1)*(x0-x2)*(x0-x3)*(x0-x4)*(x0-x6)*(x0-x7)/((x5-x1).*(x5-x2).*(x5-x3).*(x5-x4).*(x5-x6).*(x5-x7))*y5+ ...
(x0-x1)*(x0-x2)*(x0-x3)*(x0-x4)*(x0-x5)*(x0-x7)/((x6-x1).*(x6-x2).*(x6-x3).*(x6-x4).*(x6-x5).*(x6-x7))*y6+ ...
(x0-x1)*(x0-x2)*(x0-x3)*(x0-x4)*(x0-x5)*(x0-x6)/((x7-x1).*(x7-x2).*(x7-x3).*(x7-x4).*(x7-x5).*(x7-x6))*y7;
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
Here is the output of the program.
Original process average from monte carlo
MCMean =
0.264105787341005
Original process average from our simulation
ItoHermiteMean =
0.263282259095077
true Mean only applicble to standard SV mean reverting type models otherwise disregard
TrueMean =
0.263736729166551
IndexMax =
473
And here is the output graph from the new program

Here is graph from a different program run. The parameters are on the title of graph.

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