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Amin
Topic Author
Posts: 2434
Joined: July 14th, 2002, 3:00 am

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

First of all we want to calculate the differential below 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)

Before we go full-fledged towards fokker-planck, we want to solve the above integral properly. I gave a strategy to solve the above integral in previous two posts. One minor problem that can possibly affect accuracy is that the solution to above integral in not in forward coordinates at time t. We would like to make a few changes to take that into account. We write the time t term on the LHS side as
$\int_0^t d\Big[ {(w_s -\zeta_w - \int_0^s \mu(w_v) dv )}^2 \Big]$

We want to convert the above integral into another related integral as
$\int_0^t d\Big[ {(w_s -\zeta_w - \int_0^s \mu(w_v) dv )}^2 \Big]$
$=\int_0^t d\Big[ {(w_s -\zeta_w - \mu(w_s) s )}^2 \Big] + G(w_0,t)$
Both integrals are closely related in higher order terms but the advantage is that now all the values in the new integral are in forward time t coordinates plus a small value. And initial values of both integrals are exactly the same. So we could probably try a solution in the form as

$\int_0^t d\Big[ {(w_s -\zeta_w - \mu(w_s) s )}^2 \Big] + G(w_0,t) ={\sigma}^2 Z^2 \Delta t$
so we can have
${(w_t -\zeta_w - \mu(w_t) t )}^2 ={(w_t -\zeta_w )}^2 - G(w_0,t) + {\sigma}^2 Z^2 \Delta t$

Edit: This is still preliminary but I still went ahead and posted it but I expect a few more changes in a day or later tonight.

Amin
Topic Author
Posts: 2434
Joined: July 14th, 2002, 3:00 am

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

In the fokker-planck derivatives equations, drift and dw terms have identical derivatives. Writing just the relevant terms

$dw(Z \frac{\partial w}{\partial Z} + {(\frac{\partial w}{\partial Z})}^3 \frac{\partial^2 Z}{\partial w^2}) + \mu(w(t)) dt (Z \frac{\partial w}{\partial Z} - {(\frac{\partial w}{\partial Z})}^3 \frac{\partial^2 Z}{\partial w^2})$

I just wanted to comment about the second derivative term and omit the first derivative term with $Z \frac{\partial w}{\partial Z}$
$dw({(\frac{\partial w}{\partial Z})}^3 \frac{\partial^2 Z}{\partial w^2})+ \mu(w(t)) dt ( - {(\frac{\partial w}{\partial Z})}^3 \frac{\partial^2 Z}{\partial w^2})$

The above term probably has to be seen as

$\big[ dw- \mu(w(t)) dt \big]({(\frac{\partial w}{\partial Z})}^3 \frac{\partial^2 Z}{\partial w^2})$

from the SDE we know that $\big[ dw- \mu(w(t)) dt \big]=\sigma dZ(t)$

So the above derivative has to be considered as
$\big[ dw- \mu(w(t)) dt \big]({(\frac{\partial w}{\partial Z})}^3 \frac{\partial^2 Z}{\partial w^2})$
$=\big[ \sigma dZ(t)]({(\frac{\partial w}{\partial Z})}^3 \frac{\partial^2 Z}{\partial w^2})$
$=\big[ \sigma dZ(t)](- \frac{\partial^2 w}{\partial Z^2})$

I am still not completely sure how the above derivative will exactly appear in Ito-hermite equations but I tried adding

$=\big[.25 {\sigma}^2 (Z^2-1) dt](- \frac{\partial^2 w}{\partial Z^2})$

with great results though this last equation is still not final.

Amin
Topic Author
Posts: 2434
Joined: July 14th, 2002, 3:00 am

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

Post in progress.

Amin
Topic Author
Posts: 2434
Joined: July 14th, 2002, 3:00 am

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

I am posting this preliminary program that is based on Fokker-planck derivatives ( I am still using the distance from mean and have not modified that part yet.) and since it seems to work quite well, I decided to post it. It is really not a final program and will undergo several changes in next few days. But as it is a good program based on FP equation derivatives and gives a very good mean in most cases without mean correction and has excellent match to density, I am posting it for friends.
function [TrueMean,ItoHermiteMean,MCMean] = ItoHermiteWilmottMeanCorrected080FP08(x0,theta,kappa,gamma,sigma0,T)

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

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

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

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

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

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

w(1:Nn)=x0^(1-gamma)/(1-gamma);

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

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

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

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

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

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

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

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

for k = 0 : (OrderA)
for m = 0:k
l4 = k - m + 1;
for n = 0 : m
l3 = m - n + 1;
for j = 0:n
l2 = n - j + 1;
l1 = j + 1;
%Ft(l1,l2,l3,l4) = dtM^((l1-1) + (l2-1) + (l3-1) + .5* (l4-1));
Fp(l1,l2,l3,l4) = (alpha + (l1-1) * beta1 + (l2-1) * beta2 + (l3-1) * 2* gamma + (l4-1) * gamma ...
- (l1-1) - (l2-1) - 2* (l3-1) - (l4-1));
Fp1(l1,l2,l3,l4) = (alpha1 + (l1-1) * beta1 + (l2-1) * beta2 + (l3-1) * 2* gamma + (l4-1) * gamma ...
- (l1-1) - (l2-1) - 2* (l3-1) - (l4-1));

YCoeff0(l1,l2,l3,l4) =YCoeff(l1,l2,l3,l4).*mu11(l1).*mu22(l2).*sigma22(l3).*sigma11(l4);
YqCoeff0(l1,l2,l3,l4) =YqCoeff(l1,l2,l3,l4).*mu11(l1).*mu22(l2).*sigma22(l3).*sigma11(l4);
end
end
end
end

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

for tt=1:Tt

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

[wMu0dt,dwMu0dtdw,d2wMu0dtdw2,d3wMu0dtdw3,c1,c2,c3,c4] = CalculateDriftAndVolOrder4(w,wnStart,Nn,YqCoeff0,Fp1,gamma,dt);

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

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

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

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

[dwdZ,d2wdZ2A,d3wdZ3,d4wdZ4] = First4Derivatives2ndOrderEqSpaced(wnStart,Nn,dNn,w,Z);
dZdw(wnStart:Nn)=1.0./dwdZ(wnStart:Nn);

if(tt>1)

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));

u0(wnStart:Nn)=wMeanPrev;

w(wnStart:Nn)=u0(wnStart:Nn)+wMu0dt(wnStart:Nn)+ ...
sign(w(wnStart:Nn)-u0(wnStart:Nn)+dw(wnStart:Nn)+2.*sqrt(dt)*signT1(wnStart:Nn).*T1(wnStart:Nn)).* ...
sqrt(abs(sign(w(wnStart:Nn)-u0(wnStart:Nn)).*(w(wnStart:Nn)-u0(wnStart:Nn)).^2+ ...
sign(dw(wnStart:Nn)).*dw2(wnStart:Nn)+ ...
4*dt*signT1(wnStart:Nn).*T1(wnStart:Nn).^2))+ ...
4*dt*dwMu0dtdw(wnStart:Nn).*dwdZ(wnStart:Nn).^2- ...
.25.*d2wdZ2(wnStart:Nn).*sigma0.^2.*dt.*(Z(wnStart:Nn).^2-1);

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=1.0;
D1wl=(11/6*w(NnMidl-2)-3*w(NnMidl-3)+3/2*w(NnMidl-4)-1/3*w(NnMidl-5))/dNn;
D2wl=(35/12*w(NnMidl-2) -26/3*w(NnMidl-3)+ 19/2*w(NnMidl-4)-14/3*w(NnMidl-5)+11/12*w(NnMidl-6))/dNn.^2;

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

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

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

%Uncomment for fourth order monte carlo

%
%   YY(1:paths)=YY(1:paths) + ...
%       (YCoeff0(1,1,2,1).*YY(1:paths).^Fp(1,1,2,1)+YCoeff0(1,2,1,1).*YY(1:paths).^Fp(1,2,1,1)+ ...
%       YCoeff0(2,1,1,1).*YY(1:paths).^Fp(2,1,1,1))*dtM + ...
%       (YCoeff0(1,1,3,1).*YY(1:paths).^Fp(1,1,3,1)+YCoeff0(1,2,2,1).*YY(1:paths).^Fp(1,2,2,1)+ ...
%       YCoeff0(2,1,2,1).*YY(1:paths).^Fp(2,1,2,1)+YCoeff0(1,3,1,1).*YY(1:paths).^Fp(1,3,1,1)+ ...
%       YCoeff0(2,2,1,1).*YY(1:paths).^Fp(2,2,1,1)+YCoeff0(3,1,1,1).*YY(1:paths).^Fp(3,1,1,1))*dtM^2 + ...
%       (YCoeff0(1,1,4,1).*YY(1:paths).^Fp(1,1,4,1)+YCoeff0(1,2,3,1).*YY(1:paths).^Fp(1,2,3,1)+ ...
%       YCoeff0(2,1,3,1).*YY(1:paths).^Fp(2,1,3,1)+YCoeff0(1,3,2,1).*YY(1:paths).^Fp(1,3,2,1)+ ...
%       YCoeff0(2,2,2,1).*YY(1:paths).^Fp(2,2,2,1)+YCoeff0(3,1,2,1).*YY(1:paths).^Fp(3,1,2,1)+ ...
%       YCoeff0(1,4,1,1).*YY(1:paths).^Fp(1,4,1,1)+YCoeff0(2,3,1,1).*YY(1:paths).^Fp(2,3,1,1)+ ...
%       YCoeff0(3,2,1,1).*YY(1:paths).^Fp(3,2,1,1)+YCoeff0(4,1,1,1).*YY(1:paths).^Fp(4,1,1,1))*dtM^3 + ...
%        (YCoeff0(1,1,5,1).*YY(1:paths).^Fp(1,1,5,1)+YCoeff0(1,2,4,1).*yy(1:paths).^Fp(1,2,4,1)+ ...
%     YCoeff0(2,1,4,1).*YY(1:paths).^Fp(2,1,4,1)+YCoeff0(2,1,1,1).*yy(1:paths).^Fp(2,1,1,1)+ ...
%      YCoeff0(2,2,3,1).*YY(1:paths).^Fp(2,2,3,1)+ ...
%      YCoeff0(3,1,3,1).*YY(1:paths).^Fp(3,1,3,1)+YCoeff0(1,4,2,1).*YY(1:paths).^Fp(1,4,2,1)+ ...
%      YCoeff0(2,3,2,1).*YY(1:paths).^Fp(2,3,2,1)+YCoeff0(3,2,2,1).*YY(1:paths).^Fp(3,2,2,1)+ ...
%      YCoeff0(4,1,2,1).*YY(1:paths).^Fp(4,1,2,1)+YCoeff0(1,5,1,1).*YY(1:paths).^Fp(1,5,1,1)+ ...
%      YCoeff0(2,4,1,1).*YY(1:paths).^Fp(2,4,1,1)+ ...
%      YCoeff0(3,3,1,1).*YY(1:paths).^Fp(3,3,1,1)+YCoeff0(4,2,1,1).*YY(1:paths).^Fp(4,2,1,1)+ ...
%      YCoeff0(5,1,1,1).*YY(1:paths).^Fp(5,1,1,1))*dtM^4+ ...
%       ((YCoeff0(1,1,1,2).*YY(1:paths).^Fp(1,1,1,2).*sqrt(dtM))+ ...
%     (YCoeff0(1,1,2,2).*YY(1:paths).^Fp(1,1,2,2)+YCoeff0(1,2,1,2).*YY(1:paths).^Fp(1,2,1,2)+ ...
%     YCoeff0(2,1,1,2).*YY(1:paths).^Fp(2,1,1,2)).*dtM^1.5+ ...
%     (YCoeff0(1,1,3,2).*YY(1:paths).^Fp(1,1,3,2)+YCoeff0(1,2,2,2).*YY(1:paths).^Fp(1,2,2,2)+ ...
%     YCoeff0(2,1,2,2).*YY(1:paths).^Fp(2,1,2,2)+YCoeff0(1,3,1,2).*YY(1:paths).^Fp(1,3,1,2)+ ...
%     YCoeff0(2,2,1,2).*YY(1:paths).^Fp(2,2,1,2)+YCoeff0(3,1,1,2).*YY(1:paths).^Fp(3,1,1,2)).*dtM^2.5+ ...
%     (YCoeff0(1,1,4,2).*YY(1:paths).^Fp(1,1,4,2)+YCoeff0(1,2,3,2).*YY(1:paths).^Fp(1,2,3,2)+ ...
%     YCoeff0(2,1,3,2).*YY(1:paths).^Fp(2,1,3,2)+YCoeff0(1,3,2,2).*YY(1:paths).^Fp(1,3,2,2)+ ...
%     YCoeff0(2,2,2,2).*YY(1:paths).^Fp(2,2,2,2)+ YCoeff0(3,1,2,2).*YY(1:paths).^Fp(3,1,2,2)+ ...
%     YCoeff0(1,4,1,2).*YY(1:paths).^Fp(1,4,1,2)+YCoeff0(2,3,1,2).*YY(1:paths).^Fp(2,3,1,2)+ ...
%     YCoeff0(3,2,1,2).*YY(1:paths).^Fp(3,2,1,2)+YCoeff0(4,1,1,2).*YY(1:paths).^Fp(4,1,1,2)).*dtM^3.5) .*HermiteP1(2,1:paths) + ...
%     ((YCoeff0(1,1,1,3).*YY(1:paths).^Fp(1,1,1,3) *dtM) + ...
%     (YCoeff0(1,1,2,3).*YY(1:paths).^Fp(1,1,2,3)+YCoeff0(1,2,1,3).*YY(1:paths).^Fp(1,2,1,3)+ ...
%     YCoeff0(2,1,1,3).*YY(1:paths).^Fp(2,1,1,3)).*dtM^2+ ...
%     (YCoeff0(1,1,3,3).*YY(1:paths).^Fp(1,1,3,3)+YCoeff0(1,2,2,3).*YY(1:paths).^Fp(1,2,2,3)+ ...
%     YCoeff0(2,1,2,3).*YY(1:paths).^Fp(2,1,2,3) + YCoeff0(1,3,1,3).*YY(1:paths).^Fp(1,3,1,3)+ ...
%     YCoeff0(2,2,1,3).*YY(1:paths).^Fp(2,2,1,3)+YCoeff0(3,1,1,3).*YY(1:paths).^Fp(3,1,1,3)).*dtM^3).*HermiteP1(3,1:paths) + ...
%     ((YCoeff0(1,1,1,4).*YY(1:paths).^Fp(1,1,1,4)*dtM^1.5 )+ ...
%     (YCoeff0(1,1,2,4).*YY(1:paths).^Fp(1,1,2,4)+YCoeff0(1,2,1,4).*YY(1:paths).^Fp(1,2,1,4)+ ...
%     YCoeff0(2,1,1,4).*YY(1:paths).^Fp(2,1,1,4))*dtM^2.5).*HermiteP1(4,1:paths) + ...
%     (YCoeff0(1,1,1,5).*YY(1:paths).^Fp(1,1,1,5)*dtM^2.0).*HermiteP1(5,1:paths);
%

YYMean(tt)=YYMean(tt)+sum(YY(1:paths))/paths;
end
YY(YY<0)=0;
disp('Original process average from monte carlo');
MCMean=sum(YY(:))/paths %origianl coordinates monte carlo average.
disp('Original process average from our simulation');
ItoHermiteMean=sum(y_w(wnStart+1:Nn-1).*ZProb(wnStart+1:Nn-1)) %Original process average from coordinates

disp('true Mean only applicble to standard SV mean reverting type models otherwise disregard');
TrueMean=theta+(x0-theta)*exp(-kappa*dt*Tt)%Mean reverting SDE original variable true average

MaxCutOff=30;
NoOfBins=round(500*gamma^2*2*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



when you will run the program by using the parameters as

ItoHermiteWilmottMeanCorrected080FP08(1,1,1,.75,1,2)

you will get the result as

ItoHermiteMean = 1.001290326250597

Amin
Topic Author
Posts: 2434
Joined: July 14th, 2002, 3:00 am

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

First of all we want to calculate the differential below 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)

Before we go full-fledged towards fokker-planck, we want to solve the above integral properly. I gave a strategy to solve the above integral in previous two posts. One minor problem that can possibly affect accuracy is that the solution to above integral in not in forward coordinates at time t. We would like to make a few changes to take that into account. We write the time t term on the LHS side as
$\int_0^t d\Big[ {(w_s -\zeta_w - \int_0^s \mu(w_v) dv )}^2 \Big]$

We want to convert the above integral into another related integral as
$\int_0^t d\Big[ {(w_s -\zeta_w - \int_0^s \mu(w_v) dv )}^2 \Big]$
$=\int_0^t d\Big[ {(w_s -\zeta_w - \mu(w_s) s )}^2 \Big] + G(w_0,t)$
Both integrals are closely related in higher order terms but the advantage is that now all the values in the new integral are in forward time t coordinates plus a small value. And initial values of both integrals are exactly the same. So we could probably try a solution in the form as

$\int_0^t d\Big[ {(w_s -\zeta_w - \mu(w_s) s )}^2 \Big] + G(w_0,t) ={\sigma}^2 Z^2 \Delta t$
so we can have
${(w_t -\zeta_w - \mu(w_t) t )}^2 ={(w_t -\zeta_w )}^2 - G(w_0,t) + {\sigma}^2 Z^2 \Delta t$

Edit: This is still preliminary but I still went ahead and posted it but I expect a few more changes in a day or later tonight.
I copied the above post for reference and notation.
I still have not carefully verified this but had the idea that may be we can calculate analytical derivatives $\frac{\partial w}{\partial Z}$ (and higher order derivatives) of w(t) by differentiating the squared form of equation (and its other variants)

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

possibly as

$\frac{\partial}{\partial Z}\Big[{(w_t -\zeta_w - \int_0^t \mu(w_v) dv )}^2 \Big] = \frac{\partial}{\partial Z}\Big[{(w_0 -\zeta_w )}^2 \Big]$
$+ \frac{\partial}{\partial Z}\Big[{\sigma}^2 Z^2 \Delta t \Big]$

and then solving for $\frac{\partial w_t}{\partial Z}$
I am really not sure but it may possibly lead to accurate analytical versions of second and third order derivatives.

Amin
Topic Author
Posts: 2434
Joined: July 14th, 2002, 3:00 am

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

Friends I was detained in a psychiatric facility for past one month and was discharged today. Though I had been trying to not do any mathematics in the days before I was detained since I was afraid that they would possibly block my neurotransmitters for mathematics if they get to know them. However when I was detained, I tried to think of some mathematical problems very briefly without giving any special time to them. I did not have my computer on me either so I could not do any advanced work and just had some basic foresight about the problems. I will give details of my detention later in my post on the persecution blog.

I am sharing these ideas for friends who have similar research interests and would want them to freely pursue the ideas (as these ideas require a lot of work before getting into the form of appropriate algorithms) and develop them but would want to request them to make their research public once they achieve success in their research effort.

First I will want to give an idea about what I mean by Ito-Taylor solution of SDEs. Basically this is the technique I used for monte carlo simulation of the SDEs but many times Ito-Taylor based analytic solutions could be valid for several years for simpler diffusions with one drift term as I had earlier shown in this thread with a matlab program. For more involved diffusions with two or more drift terms like the ones we encounter in mean reverting stochastic volatility models, we could still have analytic solutions out to half a year or even more especially when we use time exponential based version of the mean reverting stochastic volatility equations that have more accuracy when properly expanded.

I was thinking that for the solution of Fokker-planck equation, we could use an Ito-Taylor like expression consisting of hermite polynomials of suitable order with suitable unknown coefficients and force this polynomial into FP equation and then use something like method of undetermined coefficients to determine the true value of unknown coefficients that satisfy the FP equation. Of course this is easier said than done and friends would have to make a lot of interesting modifications and changes but I was still thinking that this could possibly be a way to solve the FP equation given that Ito-Taylor solution like expression should actually be compatible with solution of FP equation after the probability part has already been removed from the FP equation as we earlier discussed in this thread.

Another idea was about finding out a proper density once Ito-Taylor solution has been derived for a limited time(not necessarily by FP equation but as we have done for monte carlo simulations). Interesting part is that this density is valid for both dz-integrals and dt-integrals. Suppose some Ito-Taylor solution is given as
$X(T)=f(Z,T)$   where Z is a standard normal and Z(t) is standard brownian motion in explanation below.
Suppose we can solve the above high order polynomial for Z and get a solution expression in the form
$Z=g(X,T)$
The above step would require finding roots of the high order polynomial and possibly different roots might be valid for different ranges to give a continuous solution of the above last equation and these complications would have to be properly worked out.
Then if the density of standard normal is denoted by $p(Z)$, the density of diffusion X could be written as
$p(g(X,T))$  where we would have to use appropriate scaling for $\frac{dZ}{dX}$
We can easily do it for density of a simple dt-integrals $X(T)=\int_0^T Z(t)dt$ by first finding the Ito-Taylor expansion of  $\int_0^T Z(t)dt$ which is given as
$X(T)=\int_0^T Z(t)dt=(1-1/\sqrt{3})[Z T^{1.5}]$ giving us the solution
$Z=\frac{X(T)}{((1-1/\sqrt{3})T^{1.5})}$
which when substituted in p(Z) would give us the density solution for X(T) as
$p(X)=\frac{1}{(1-1/\sqrt{3})\sqrt{2 \pi T^3}}exp(-.5 {(\frac{X}{(1-1/\sqrt{3})T^{1.5}})}^2)$
Similar technique can surely be used for far more difficult diffusions, dt-integrals and dz-integrals but of course other considerations like appropriate root-finding that gives a continuous solution would have to be properly worked out,

Another idea was for Ito-Taylor solution mean that could be used to project the mean of the SDEs. For example in transition probabilities framework, we found the mean by projecting mean at each node and then weighting each mean with probability mass at each node and then adding all of the weighted means. But in Ito-Taylor expansion, all the hermite polynomials with different coefficients have zero mean and we need to consider only the pure drift and pure quadratic variation parts and concentrating on pure mean and pure quadratic variations could give us better accuracy and some more research could give us ideas where to truncate the quadratic variation part especially close to zero for accuracy. Removing the zero mean hermite polynomials would give us better accuracy since usually our grid would be sparse and non-symmetric where including these hermite polynomials in our mean calculations could add to inaccuracies.

Another idea I was thinking was about addition of densities which is done basically by convolution of two densities. Since solution expression for Ito-Taylor form is already known in terms of weighted sum of hermite polynomials, we could probably find a way where we could pre-solve the expressions for convolution of hermite polynomial based part of the densities(which is fixed for a certain polynomial order of both densities) and then just multiply them with appropriate weights to find the summation density. But this will have to be properly worked out.

Another idea was about inference of SDEs. When we have some data where we want to fit an SDE, we could throw a trial SDE and solve for the value of standard normal random number that fits the data and the SDE. And then make a density out of all the data and trial SDE fitted standard normal random numbers. The closest this fitted standard normal density is to the true standard normal density, the better the fit of the trial SDE would be to the data. For example if we do monte carlo with a known SDE and generate data, our fitted standard normal would be the same as the driving standard normal which are already known at simulation time and when we plot those driving standard normals, they will give us a true standard normal distribution since true distribution is identical with the already known driving SDE. This is basically an optimization problem in which we successively throw trial SDEs so that our trail SDE and data-fitted standard normals approach a true standard normal distribution.

I had these ideas while in detention and mind control guys were not happy when I had these these ideas and I seriously think that my thinking about these ideas prolonged my detention.

Amin
Topic Author
Posts: 2434
Joined: July 14th, 2002, 3:00 am

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

Removing the zero mean hermite polynomials would give us better accuracy since usually our grid would be sparse and non-symmetric where including these hermite polynomials in our mean calculations could add to inaccuracies
Sorry for bad use of the word sparse. What I basically meant was that the transition probabilities grid  would usually have large spacing.

Amin
Topic Author
Posts: 2434
Joined: July 14th, 2002, 3:00 am

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

Friends, I have been playing with my Ito-hermite version of SDE density evolution program. Earlier programs were good with the main body of the distribution but off in the right tail even with mean correction. But I have been able to make some changes to the earlier algorithm so it works great even deep in the tail when mean correction is used. I have been trying the program with the same mean reverting stochastic volatility type equations yet but I am sure it will work great with general stochastic differential equations when mean correction could be used. I will be posting the matlab program later today or probably tomorrow after making a few more minor changes.

Amin
Topic Author
Posts: 2434
Joined: July 14th, 2002, 3:00 am

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

I am posting a few graphs of mean reverting SV type SDE densities with my new version of SDE density evolution program. I have posted a graph for each SDE density focusing on the main body of the SDE density followed by another graph focusing on the tail of the same SDE density. SDE parameters are shown on the title of the graph. Tail is very well captured by this new program but it is probably 98% perfect but still extremely good. I have used density mean-correction technique for these densities. I will be posting the matlab program in a few hours. Here are the graphs.

Amin
Topic Author
Posts: 2434
Joined: July 14th, 2002, 3:00 am

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

Here I am posting the new improved program that I used to create the above graphs. I continued to change NoOfBins parameter used for graphing. Here is the new program.
function [TrueMean,ItoHermiteMean,MCMean] = ItoHermiteWilmottMeanCorrected04WNew02(x0,theta,kappa,gamma,sigma0,T)

%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.

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

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

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

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

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

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

w(1:Nn)=x0^(1-gamma)/(1-gamma);

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

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

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

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

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

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

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

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

for k = 0 : (OrderA)
for m = 0:k
l4 = k - m + 1;
for n = 0 : m
l3 = m - n + 1;
for j = 0:n
l2 = n - j + 1;
l1 = j + 1;
%Ft(l1,l2,l3,l4) = dtM^((l1-1) + (l2-1) + (l3-1) + .5* (l4-1));
Fp(l1,l2,l3,l4) = (alpha + (l1-1) * beta1 + (l2-1) * beta2 + (l3-1) * 2* gamma + (l4-1) * gamma ...
- (l1-1) - (l2-1) - 2* (l3-1) - (l4-1));
Fp1(l1,l2,l3,l4) = (alpha1 + (l1-1) * beta1 + (l2-1) * beta2 + (l3-1) * 2* gamma + (l4-1) * gamma ...
- (l1-1) - (l2-1) - 2* (l3-1) - (l4-1));

YCoeff0(l1,l2,l3,l4) =YCoeff(l1,l2,l3,l4).*mu11(l1).*mu22(l2).*sigma22(l3).*sigma11(l4);
YqCoeff0(l1,l2,l3,l4) =YqCoeff(l1,l2,l3,l4).*mu11(l1).*mu22(l2).*sigma22(l3).*sigma11(l4);
end
end
end
end

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

for tt=1:Tt

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

[wMu0dt,dwMu0dtdw,d2wMu0dtdw2,d3wMu0dtdw3,c1,c2,c3,c4] = CalculateDriftAndVolOrder4(w,wnStart,Nn,YqCoeff0,Fp1,gamma,dt);

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

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

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

[dwdZ,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)=(.5*d2wMu0dtdw2(wnStart:Nn).*dwdZ(wnStart:Nn).^2 - ...
%           1*dwMu0dtdw(wnStart:Nn).*d2wdZ2(wnStart:Nn));
% Correction1(wnStart:Nn)=-sqrt(2/3)/(22/7)^2.*CTerm(wnStart:Nn).*H2(wnStart:Nn);
% Correction0(wnStart:Nn)=-sqrt(2/3)/(22/7)^2.*(d2wdZ2(wnStart:Nn)-.25*dwdZ(wnStart:Nn).^2).*H2(wnStart:Nn);

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);

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;

%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));

u0(wnStart:Nn)=wMeanPrev;

w(wnStart:Nn)=u0(wnStart:Nn)+wMu0dt(wnStart:Nn)+ ...
sign(w(wnStart:Nn)-u0(wnStart:Nn)+dw(wnStart:Nn)+sqrt(3).*sqrt(dt)*signT1(wnStart:Nn).*T1(wnStart:Nn)).* ...
sqrt(abs(sign(w(wnStart:Nn)-u0(wnStart:Nn)).*(w(wnStart:Nn)-u0(wnStart:Nn)).^2+ ...
sign(dw(wnStart:Nn)).*dw2(wnStart:Nn)+ ...
3*dt*signT1(wnStart:Nn).*T1(wnStart:Nn).^2))+ ...
4*dt*dwMu0dtdw(wnStart:Nn).*dwdZ(wnStart:Nn).^2.*Z(wnStart:Nn)+ ...
+dt *Correction0(wnStart:Nn).*exp(-.5*(1-gamma)*sigma0.^2.*Z(wnStart:Nn).^2) + ...
Correction1(wnStart:Nn).*exp(-.5*(1-gamma)*sigma0.^2.*Z(wnStart:Nn).^2);

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=1.0;
D1wl=(11/6*w(NnMidl-2)-3*w(NnMidl-3)+3/2*w(NnMidl-4)-1/3*w(NnMidl-5))/dNn;
D2wl=(35/12*w(NnMidl-2) -26/3*w(NnMidl-3)+ 19/2*w(NnMidl-4)-14/3*w(NnMidl-5)+11/12*w(NnMidl-6))/dNn.^2;

%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);

%The above mean correction is only valid for
%standard mean reverting stochastic volatility type SDEs.

end

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

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

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

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

%Uncomment for fourth order monte carlo

%
%   YY(1:paths)=YY(1:paths) + ...
%       (YCoeff0(1,1,2,1).*YY(1:paths).^Fp(1,1,2,1)+YCoeff0(1,2,1,1).*YY(1:paths).^Fp(1,2,1,1)+ ...
%       YCoeff0(2,1,1,1).*YY(1:paths).^Fp(2,1,1,1))*dtM + ...
%       (YCoeff0(1,1,3,1).*YY(1:paths).^Fp(1,1,3,1)+YCoeff0(1,2,2,1).*YY(1:paths).^Fp(1,2,2,1)+ ...
%       YCoeff0(2,1,2,1).*YY(1:paths).^Fp(2,1,2,1)+YCoeff0(1,3,1,1).*YY(1:paths).^Fp(1,3,1,1)+ ...
%       YCoeff0(2,2,1,1).*YY(1:paths).^Fp(2,2,1,1)+YCoeff0(3,1,1,1).*YY(1:paths).^Fp(3,1,1,1))*dtM^2 + ...
%       (YCoeff0(1,1,4,1).*YY(1:paths).^Fp(1,1,4,1)+YCoeff0(1,2,3,1).*YY(1:paths).^Fp(1,2,3,1)+ ...
%       YCoeff0(2,1,3,1).*YY(1:paths).^Fp(2,1,3,1)+YCoeff0(1,3,2,1).*YY(1:paths).^Fp(1,3,2,1)+ ...
%       YCoeff0(2,2,2,1).*YY(1:paths).^Fp(2,2,2,1)+YCoeff0(3,1,2,1).*YY(1:paths).^Fp(3,1,2,1)+ ...
%       YCoeff0(1,4,1,1).*YY(1:paths).^Fp(1,4,1,1)+YCoeff0(2,3,1,1).*YY(1:paths).^Fp(2,3,1,1)+ ...
%       YCoeff0(3,2,1,1).*YY(1:paths).^Fp(3,2,1,1)+YCoeff0(4,1,1,1).*YY(1:paths).^Fp(4,1,1,1))*dtM^3 + ...
%        (YCoeff0(1,1,5,1).*YY(1:paths).^Fp(1,1,5,1)+YCoeff0(1,2,4,1).*yy(1:paths).^Fp(1,2,4,1)+ ...
%     YCoeff0(2,1,4,1).*YY(1:paths).^Fp(2,1,4,1)+YCoeff0(2,1,1,1).*yy(1:paths).^Fp(2,1,1,1)+ ...
%      YCoeff0(2,2,3,1).*YY(1:paths).^Fp(2,2,3,1)+ ...
%      YCoeff0(3,1,3,1).*YY(1:paths).^Fp(3,1,3,1)+YCoeff0(1,4,2,1).*YY(1:paths).^Fp(1,4,2,1)+ ...
%      YCoeff0(2,3,2,1).*YY(1:paths).^Fp(2,3,2,1)+YCoeff0(3,2,2,1).*YY(1:paths).^Fp(3,2,2,1)+ ...
%      YCoeff0(4,1,2,1).*YY(1:paths).^Fp(4,1,2,1)+YCoeff0(1,5,1,1).*YY(1:paths).^Fp(1,5,1,1)+ ...
%      YCoeff0(2,4,1,1).*YY(1:paths).^Fp(2,4,1,1)+ ...
%      YCoeff0(3,3,1,1).*YY(1:paths).^Fp(3,3,1,1)+YCoeff0(4,2,1,1).*YY(1:paths).^Fp(4,2,1,1)+ ...
%      YCoeff0(5,1,1,1).*YY(1:paths).^Fp(5,1,1,1))*dtM^4+ ...
%       ((YCoeff0(1,1,1,2).*YY(1:paths).^Fp(1,1,1,2).*sqrt(dtM))+ ...
%     (YCoeff0(1,1,2,2).*YY(1:paths).^Fp(1,1,2,2)+YCoeff0(1,2,1,2).*YY(1:paths).^Fp(1,2,1,2)+ ...
%     YCoeff0(2,1,1,2).*YY(1:paths).^Fp(2,1,1,2)).*dtM^1.5+ ...
%     (YCoeff0(1,1,3,2).*YY(1:paths).^Fp(1,1,3,2)+YCoeff0(1,2,2,2).*YY(1:paths).^Fp(1,2,2,2)+ ...
%     YCoeff0(2,1,2,2).*YY(1:paths).^Fp(2,1,2,2)+YCoeff0(1,3,1,2).*YY(1:paths).^Fp(1,3,1,2)+ ...
%     YCoeff0(2,2,1,2).*YY(1:paths).^Fp(2,2,1,2)+YCoeff0(3,1,1,2).*YY(1:paths).^Fp(3,1,1,2)).*dtM^2.5+ ...
%     (YCoeff0(1,1,4,2).*YY(1:paths).^Fp(1,1,4,2)+YCoeff0(1,2,3,2).*YY(1:paths).^Fp(1,2,3,2)+ ...
%     YCoeff0(2,1,3,2).*YY(1:paths).^Fp(2,1,3,2)+YCoeff0(1,3,2,2).*YY(1:paths).^Fp(1,3,2,2)+ ...
%     YCoeff0(2,2,2,2).*YY(1:paths).^Fp(2,2,2,2)+ YCoeff0(3,1,2,2).*YY(1:paths).^Fp(3,1,2,2)+ ...
%     YCoeff0(1,4,1,2).*YY(1:paths).^Fp(1,4,1,2)+YCoeff0(2,3,1,2).*YY(1:paths).^Fp(2,3,1,2)+ ...
%     YCoeff0(3,2,1,2).*YY(1:paths).^Fp(3,2,1,2)+YCoeff0(4,1,1,2).*YY(1:paths).^Fp(4,1,1,2)).*dtM^3.5) .*HermiteP1(2,1:paths) + ...
%     ((YCoeff0(1,1,1,3).*YY(1:paths).^Fp(1,1,1,3) *dtM) + ...
%     (YCoeff0(1,1,2,3).*YY(1:paths).^Fp(1,1,2,3)+YCoeff0(1,2,1,3).*YY(1:paths).^Fp(1,2,1,3)+ ...
%     YCoeff0(2,1,1,3).*YY(1:paths).^Fp(2,1,1,3)).*dtM^2+ ...
%     (YCoeff0(1,1,3,3).*YY(1:paths).^Fp(1,1,3,3)+YCoeff0(1,2,2,3).*YY(1:paths).^Fp(1,2,2,3)+ ...
%     YCoeff0(2,1,2,3).*YY(1:paths).^Fp(2,1,2,3) + YCoeff0(1,3,1,3).*YY(1:paths).^Fp(1,3,1,3)+ ...
%     YCoeff0(2,2,1,3).*YY(1:paths).^Fp(2,2,1,3)+YCoeff0(3,1,1,3).*YY(1:paths).^Fp(3,1,1,3)).*dtM^3).*HermiteP1(3,1:paths) + ...
%     ((YCoeff0(1,1,1,4).*YY(1:paths).^Fp(1,1,1,4)*dtM^1.5 )+ ...
%     (YCoeff0(1,1,2,4).*YY(1:paths).^Fp(1,1,2,4)+YCoeff0(1,2,1,4).*YY(1:paths).^Fp(1,2,1,4)+ ...
%     YCoeff0(2,1,1,4).*YY(1:paths).^Fp(2,1,1,4))*dtM^2.5).*HermiteP1(4,1:paths) + ...
%     (YCoeff0(1,1,1,5).*YY(1:paths).^Fp(1,1,1,5)*dtM^2.0).*HermiteP1(5,1:paths);
%

YYMean(tt)=YYMean(tt)+sum(YY(1:paths))/paths;
end
YY(YY<0)=0;
disp('Original process average from monte carlo');
MCMean=sum(YY(:))/paths %origianl coordinates monte carlo average.
disp('Original process average from our simulation');
ItoHermiteMean=sum(y_w(wnStart+1:Nn-1).*ZProb(wnStart+1:Nn-1)) %Original process average from coordinates

disp('true Mean only applicble to standard SV mean reverting type models otherwise disregard');
TrueMean=theta+(x0-theta)*exp(-kappa*dt*Tt)%Mean reverting SDE original variable true average

MaxCutOff=30;
NoOfBins=round(1000*gamma^2*4*sigma0);
[YDensity,IndexOutY,IndexMaxY] = MakeDensityFromSimulation_Infiniti_NEW(YY,paths,NoOfBins,MaxCutOff );
plot(y_w(wnStart+1:Nn-1),fy_w(wnStart+1:Nn-1),'r',IndexOutY(1:IndexMaxY),YDensity(1:IndexMaxY),'g');

title(sprintf('Ito-Hermite Method VS Monte Carlo Density, x0 = %.3f,theta=%.3f,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));
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
The above program when run with the following line of code(these are parameters of the last graph.)
ItoHermiteWilmottMeanCorrected04WNew02(.1,.05,1.0,.75,.7,2.0)
will result in following last output:
Original process average from monte carlo
MCMean =
0.056659206709890
Original process average from our simulation
ItoHermiteMean =
0.056762486183782
true Mean only applicble to standard SV mean reverting type models otherwise disregard
TrueMean =
0.056766764161831
IndexMax =
1576
red line is density of SDE from Ito-Hermite method, green is monte carlo.
ans =
0.056766764161831

Amin
Topic Author
Posts: 2434
Joined: July 14th, 2002, 3:00 am

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

Please replace the terms in posted program with these new terms:

CTerm(wnStart:Nn)=(.25*d2wMu0dtdw2(wnStart:Nn).*dwdZ(wnStart:Nn).^1.*Z(wnStart:Nn)/sqrt(2.0) - ...
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).^1.*Z(wnStart:Nn)/sqrt(2.0)).*H2(wnStart:Nn);

Edit: Please just disregard this post. I will come back with more details later.

Amin
Topic Author
Posts: 2434
Joined: July 14th, 2002, 3:00 am

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

I am posting a different program which is also very good. But wonderful thing is that in this program most derivatives are almost the same as the ones analytically calculated as in post 920. I have added extra multiplication with second or first hermite polynomial and scaling constant is still arbitrary but body of derivatives terms(apart from extra multiplication with hermite polynomials and scaling and another exponential multiplication) are exactly as different terms in analytic solution of FP equation as in post 920. It is still not a final program but I decided to share it.
function [TrueMean,ItoHermiteMean,y_w,fy_w,wnStart] = ItoHermiteWilmottMeanCorrected080FP08WA02F(x0,theta,kappa,gamma,sigma0,T)

%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*T;%16*16;     % Number of simulation levels. Terminal time= Tt*dt; //.125/32*32*16=2 year;
OrderA=4;  %
OrderM=4;  %
%dtM=.125/8;%Monte carlo time interval size dtM.
%TtM=8*8;%Monte carlo number of simulation intervals.
dtM=dt*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=.25;   % starting value of SDE
beta1=0.0;
beta2=1.0;   % Second drift term power.
%gamma=.95;%50;   % volatility power.
%kappa=.5;   %mean reversion parameter.
%theta=.075;%mean reversion target

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

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

w(1:Nn)=x0^(1-gamma)/(1-gamma);

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

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

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

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

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

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

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

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

for k = 0 : (OrderA)
for m = 0:k
l4 = k - m + 1;
for n = 0 : m
l3 = m - n + 1;
for j = 0:n
l2 = n - j + 1;
l1 = j + 1;
%Ft(l1,l2,l3,l4) = dtM^((l1-1) + (l2-1) + (l3-1) + .5* (l4-1));
Fp(l1,l2,l3,l4) = (alpha + (l1-1) * beta1 + (l2-1) * beta2 + (l3-1) * 2* gamma + (l4-1) * gamma ...
- (l1-1) - (l2-1) - 2* (l3-1) - (l4-1));
Fp1(l1,l2,l3,l4) = (alpha1 + (l1-1) * beta1 + (l2-1) * beta2 + (l3-1) * 2* gamma + (l4-1) * gamma ...
- (l1-1) - (l2-1) - 2* (l3-1) - (l4-1));

YCoeff0(l1,l2,l3,l4) =YCoeff(l1,l2,l3,l4).*mu11(l1).*mu22(l2).*sigma22(l3).*sigma11(l4);
YqCoeff0(l1,l2,l3,l4) =YqCoeff(l1,l2,l3,l4).*mu11(l1).*mu22(l2).*sigma22(l3).*sigma11(l4);
end
end
end
end

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

for tt=1:Tt

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

[wMu0dt,dwMu0dtdw,d2wMu0dtdw2,d3wMu0dtdw3,c1,c2,c3,c4] = CalculateDriftAndVolOrder4(w,wnStart,Nn,YqCoeff0,Fp1,gamma,dt);

dw(wnStart:Nn)=c1(wnStart:Nn).*Z(wnStart:Nn) ;% ...
dw2(wnStart:Nn)=dw(wnStart:Nn).^2;
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)=(.5*d2wMu0dtdw2(wnStart:Nn).*dwdZ(wnStart:Nn).^2 - ...
%           1*dwMu0dtdw(wnStart:Nn).*d2wdZ2(wnStart:Nn));
% Correction1(wnStart:Nn)=-sqrt(2/3)/(22/7)^2.*CTerm(wnStart:Nn).*H2(wnStart:Nn);
% Correction0(wnStart:Nn)=-sqrt(2/3)/(22/7)^2.*(d2wdZ2(wnStart:Nn)-.25*dwdZ(wnStart:Nn).^2).*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)

%
%
%    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)))+ ...
%            +dt *Correction0(wnStart:Nn).*exp(-.5*(1-gamma)*sigma0.^2.*Z(wnStart:Nn).^2) + ...
%           Correction1(wnStart:Nn).*exp(-.5*(1-gamma)*sigma0.^2.*Z(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));

u0(wnStart:Nn)=wMeanPrev;

w(wnStart:Nn)=u0(wnStart:Nn)+wMu0dt(wnStart:Nn)+ ...
sign(w(wnStart:Nn)-u0(wnStart:Nn)+dw(wnStart:Nn)+0*sqrt(2).*sqrt(dt)*signT1(wnStart:Nn).*T1(wnStart:Nn)).* ...
sqrt(abs(sign(w(wnStart:Nn)-u0(wnStart:Nn)).*(w(wnStart:Nn)-u0(wnStart:Nn)).^2+ ...
sign(dw(wnStart:Nn)).*dw2(wnStart:Nn)+ ...
0*2*dt*signT1(wnStart:Nn).*T1(wnStart:Nn).^2))- ...
2*dt*dwMu0dtdw(wnStart:Nn).*dwdZ(wnStart:Nn).^2.*H2(wnStart:Nn)- ...
dt*signT1(wnStart:Nn).*T1(wnStart:Nn).^2.*Z(wnStart:Nn).^1+ ...
+dt *Correction0(wnStart:Nn).*exp(-.5*(1-gamma)*sigma0.^2.*Z(wnStart:Nn).^2) + ...
+Correction1(wnStart:Nn).*exp(-.5*(1-gamma)*sigma0.^2.*Z(wnStart:Nn).^2);

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=1.0;
D1wl=(11/6*w(NnMidl-2)-3*w(NnMidl-3)+3/2*w(NnMidl-4)-1/3*w(NnMidl-5))/dNn;
D2wl=(35/12*w(NnMidl-2) -26/3*w(NnMidl-3)+ 19/2*w(NnMidl-4)-14/3*w(NnMidl-5)+11/12*w(NnMidl-6))/dNn.^2;

%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(500*gamma^2*2*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 this program with command
ItoHermiteWilmottMeanCorrected080FP08WA02F(1,1,.7,.65,1,2).
Last Output:
MCMean =
0.995543768507918
Original process average from our simulation
ItoHermiteMean =
0.999994005675058

Here is the graph output of the program with above parameters.

Amin
Topic Author
Posts: 2434
Joined: July 14th, 2002, 3:00 am

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

I have been trying to mould the evolution equation for Fokker-planck PDE into a form that new evolution equation does not depend upon distance from the mean of the SDE distribution but rather depends upon distance from the next/adjacent grid point so we can use local derivatives for the evolution equation everywhere. I have had good success in devising the new basic framework for short time(with distances and derivatives from adjacent points) when second derivatives are not large and now I would be adding the effect of second derivatives in this new evolution equation. I will also be posting the program with simple analytics behind it by tomorrow.

Amin
Topic Author
Posts: 2434
Joined: July 14th, 2002, 3:00 am

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

In my earlier Ito-hermite evolution equations, the barebone (part of the solution equation that depended only on first derivatives without including the effect of second and higher derivatives) solution of general SDEs was borrowed from brownian motion solution equation. Brownian motion solution equation had a constant first derivative (dB/dZ) but for general SDEs, the first derivative is never constant and using brownian motion barebone created extra errors and there were problems for finding a general solution after adding the effects of second derivatives since even the first derivatives solution part was off its true values. I did some simple analysis so that we could properly capture the effect of first derivatives in the solution of Fokker-planck equation. I still have not worked on adding the effects of second order derivatives but the results of solution with first order derivatives are quite encouraging. This was a major problem in the previous attempts at solution of FPE that first order solution was off and then adding effects of second order derivatives would not be able to give us a good result.
Here I write the brownian motion based barebone solution and then describe the new correct first order bare bone solution and then later show how I derived the correct first order derivatives based barebone solution.
Brownain motion solution is given by the equations below as
$d[(B(t)-\mu_B - \mu(t)dt )^2]={\sigma}^2 Z^2 \Delta t$
from which we derived the evolution equation as
$(B(t_1)-\mu_B - \mu(t)dt )^2=(B(t_0)-\mu_B)^2 + {\sigma}^2 Z^2 \Delta t$

We based our barebone first order solution for FPE of Bessel form SDEs on the above evolution equation for brownian motion as
$(w(t_1)-\mu_w - \mu(t)dt )^2=(w(t_0)-\mu_w)^2 + {\sigma}^2 Z^2 \Delta t$

But since first order derivative is not constant, the above equation is not valid for first order derivatives effect for solution of FPE of general SDEs even when we are not accounting for the second order derivatives. I believe that appropriate equation that properly accounts for first order derivatives should be given as( I will derive it after writing the equation here)

$d[(w_n(t)-w_{n-1}(t)+Z_{n-1} \frac{dw_n}{dZ_n} - \mu(t)dt )^2]={\sigma}^2 Z^2 \Delta t$

In my matlab program, I wrote the barebone(including only the effect of first order derivatives) evolution equation based on the above equation as

$(w_n(t_1)-w_{n-1}(t_0)+Z_{n-1} \frac{dw_n}{dZ_n}(t_0) - \mu(t)dt )^2$
$=(w_n(t_0)-w_{n-1}(t_0)+Z_{n-1} \frac{dw_n}{dZ_n}(t_0))^2 + {\sigma}^2 Z^2 \Delta t$
further resulting in solution equation
$w_n(t_1)=w_{n-1}(t_0)-Z_{n-1} \frac{dw_n}{dZ_n}(t_0) + \mu(t)dt$
$+\sqrt{(w_n(t_0)-w_{n-1}(t_0)+Z_{n-1} \frac{dw_n}{dZ_n}(t_0))^2 + {\sigma}^2 Z^2 \Delta t}$
We start calculating the above solution away from median on both sides. The above equation is valid for points to the right of median and the following equation is valid for points to the left of the median.
$w_n(t_1)=w_{n+1}(t_0)-Z_{n+1} \frac{dw_n}{dZ_n}(t_0) + \mu(t)dt$
$+\sqrt{(w_n(t_0)-w_{n+1}(t_0)+Z_{n+1} \frac{dw_n}{dZ_n}(t_0))^2 + {\sigma}^2 Z^2 \Delta t}$
I have not been able to do extensive experiments but I believe that we can iteratively solve the equation and might want to take  $w_{n-1}(t_1)$ or $w_{n+1}(t_1)$instead of $w_{n-1}(t_0)$ or $w_{n+1}(t_0)$ in the first term of RHS of solution i.e use the updated value of adjacent grid point.

Here is how I solved for the above equation.
First of all we have the first derivative term in FP equation given as $Z\frac{dw}{dZ}$.  In case of constant first derivative, we used the equation
$Z \frac{dw}{dZ}=Z_n \frac{(w_n-w_0)}{Z_n}$
but for general SDE, the true Finite difference derivatives approximation equation at nth grid point should be
$Z \frac{dw}{dZ}=Z_n \frac{(w_n-w_{n-1})}{(Z_n-Z_{n-1})}$
We want to write the above true approximation in terms of sum of brownian motion approximation and another yet unknown term $T_1$ as
$Z_n \frac{(w_n-w_{n-1})}{(Z_n-Z_{n-1})}=Z_n \frac{(w_n-w_0)}{Z_n}+T_1$
we use $\Delta w=w_n-w_{n-1}$ and $\Delta Z=Z_n-Z_{n-1}$

solving for $T_1$, we get
$T_1= \frac{(-Z_n w_{n-1} + Z_{n-1} w_n + Z_n w_0 -Z_{n-1} w_0)}{\Delta Z}$
In the above, we substitute $Z_n=Z_{n-1}+\Delta Z$ and $w_n=w_{n-1}+\Delta w$
and simplify to get the solution
$T_1= -w_{n-1} + Z_{n-1} \frac{\Delta w}{\Delta Z} + w_0$

now please recall the previous equation
$Z_n \frac{(w_n-w_{n-1})}{(Z_n-Z_{n-1})}=Z_n \frac{(w_n-w_0)}{Z_n}+T_1$
and substitute the newly calculated value of $T_1$ on RHS to get

$Z \frac{dw}{dZ}=Z_n \frac{(w_n-w_{n-1})}{(Z_n-Z_{n-1})}=Z_n \frac{(w_n-w_0)}{Z_n} -w_{n-1} + Z_{n-1} \frac{\Delta w}{\Delta Z} + w_0$
$=w_n-w_{n-1}+Z_{n-1} \frac{dw}{dZ}$
which is what I have used in the barebone first derivatives based solution of the SDE density as given by FP equation.
The above calculations lead to the equations below that I have written again after copying from before as

$(w_n(t_1)-w_{n-1}(t_0)+Z_{n-1} \frac{dw_n}{dZ_n}(t_0) - \mu(t)dt )^2$
$=(w_n(t_0)-w_{n-1}(t_0)+Z_{n-1} \frac{dw_n}{dZ_n}(t_0))^2 + {\sigma}^2 Z^2 \Delta t$
further resulting in solution equation
$w_n(t_1)=w_{n-1}(t_0)-Z_{n-1} \frac{dw_n}{dZ_n}(t_0) + \mu(t)dt$
$+\sqrt{(w_n(t_0)-w_{n-1}(t_0)+Z_{n-1} \frac{dw_n}{dZ_n}(t_0))^2 + {\sigma}^2 Z^2 \Delta t}$

We still have to do some work to appropriately add the effect of second and third derivatives terms. But the above solution works very well as long as 2nd order derivative is negligible for small times or low mean reversion values or for simple CEV type noises.
I will be posting a matlab program in a little bit in the next post.

Amin
Topic Author
Posts: 2434
Joined: July 14th, 2002, 3:00 am

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

Here is the matlab program for the above barebone first order solution of FPE.
function [TrueMean,ItoHermiteMean,y_w,fy_w,wnStart] = FPERevisited02()

%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=64;%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=1;   % starting value of SDE
beta1=0.0;
beta2=1.0;   % Second drift term power.
gamma=.65;%50;   % volatility power.
kappa=.50;   %mean reversion parameter.
theta=1.0;%mean reversion target

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

mu1=+1*theta*kappa;   %first drift coefficient.
mu2=-1*kappa;    % Second drift coefficient.
%mu1=0;
%mu2=0;
sigma0=1.0;%Volatility value
alpha=1;% x^alpha is being expanded. This is currently for monte carlo only.
alpha1=1-gamma;

w(1:Nn)=x0^(1-gamma)/(1-gamma);

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

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

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

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

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

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

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

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

for k = 0 : (OrderA)
for m = 0:k
l4 = k - m + 1;
for n = 0 : m
l3 = m - n + 1;
for j = 0:n
l2 = n - j + 1;
l1 = j + 1;
%Ft(l1,l2,l3,l4) = dtM^((l1-1) + (l2-1) + (l3-1) + .5* (l4-1));
Fp(l1,l2,l3,l4) = (alpha + (l1-1) * beta1 + (l2-1) * beta2 + (l3-1) * 2* gamma + (l4-1) * gamma ...
- (l1-1) - (l2-1) - 2* (l3-1) - (l4-1));
Fp1(l1,l2,l3,l4) = (alpha1 + (l1-1) * beta1 + (l2-1) * beta2 + (l3-1) * 2* gamma + (l4-1) * gamma ...
- (l1-1) - (l2-1) - 2* (l3-1) - (l4-1));

YCoeff0(l1,l2,l3,l4) =YCoeff(l1,l2,l3,l4).*mu11(l1).*mu22(l2).*sigma22(l3).*sigma11(l4);
YqCoeff0(l1,l2,l3,l4) =YqCoeff(l1,l2,l3,l4).*mu11(l1).*mu22(l2).*sigma22(l3).*sigma11(l4);
end
end
end
end

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

for tt=1:Tt

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

[wMu0dt,dwMu0dtdw,d2wMu0dtdw2,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)=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)

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)
[w_Z0] = 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.

sign(+dw(wnStart:Nn)).*dw2(wnStart:Nn)));

end

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%Below are instability correction equations. There is instability in the
% SDE representation equation roughly between -.5 SD to +.5 SD in normal terms.
%Instability correction Algorithm takes two ends that are here given as
%lower end=w(NnMidl-2) and upper end=w(NnMidh+3).
%In the block below the interpolation starts from lower end.
%1st equation is
%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=1.0;
D1wl=(11/6*w(NnMidl-2)-3*w(NnMidl-3)+3/2*w(NnMidl-4)-1/3*w(NnMidl-5))/dNn;
D2wl=(35/12*w(NnMidl-2) -26/3*w(NnMidl-3)+ 19/2*w(NnMidl-4)-14/3*w(NnMidl-5)+11/12*w(NnMidl-6))/dNn.^2;

%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(500*gamma^2*2*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 with hard-coded parameters.

MCMean =
1.000103151236431
Original process average from our simulation
ItoHermiteMean =
0.999995633018249
true Mean only applicble to standard SV mean reverting type models otherwise disregard
TrueMean =
1
IndexMax =
424