amin


Total Posts: 264 
Joined: Aug 2005 


Here I do the complete analysis as I promised yesterday.
Equation(1)
When we are given initial values t0 and y(t0), it is obvious we can easily solve for dy/dt(y(t0),t0). Let us take a total derivative of Equation(1), it reads
Equation(2)
please notice that I missed a term yesterday when I wrote the equation (2). We make the following observation that time dependent partial derivatives in third and fourth term of equation(2) can be combined to give the expression below.
Equation(3) Now we can write the third and fourth term in equation (2) to give us the Equation(4)
using the above expression in equation 2, we get Equation(5)
which can be solved to give us a second order ODE given below. Equation(6)
which can easily be solved by the method of iterated integrals if we can integrate the expressions in corresponding integrals.
We notice that at any given time tn, equation (1) has to be satisfied as
which means that everywhere on the surface F, the following equation would have to be true as well Equation(7) which makes our second order ODE in equation (6) to be solved as Equation(8)




amin


Total Posts: 264 
Joined: Aug 2005 


Here is crossreference to another related thread in offtopic for applications of the research in this technical forum thread. https://defence.pk/pdf/threads/proposalforananotechnologyopenknowledgeresearcheffort.482885/ I thought some friends would be interested. 


amin


Total Posts: 264 
Joined: Aug 2005 

 
pj


Total Posts: 3356 
Joined: Jun 2004 


Hi Amin,
You should use this link instead. W****tt is unpronounceable name here. 
The older I grow, the more I distrust the familiar doctrine that age brings wisdom
Henry L. Mencken 

amin


Total Posts: 264 
Joined: Aug 2005 


In a new post there .https://goo.gl/ue7PfT Post:565, I have written the solution to stochastic integrals that are needed for monte carlo. Now I believe people have full recipe for higher order monte carlo 



amin


Total Posts: 264 
Joined: Aug 2005 


Here is a matlab program that simulates higher order iterated integral monte carlo of lognormal diffusion with drift and compares it with traditional Euler method and also with exact analytic solution.
function [] = SVMonteCarloHigherOrderInfiniti( ) %SDE being discretized is %dX(t)=mu0*X(t)*dt+simga0*X(t)*dz(t) T_index=8; %No. of Simulation time steps dt=.125/1.0; % Discretization spacing TT=T_index*dt;%%Terminal Time mu0=.40; %drift sigma0=1.0;%volatility X_0=1.0; %Starting Value of the lognormal diffusion paths=1000000; %No Of Paths. X(1:paths)=X_0; %Traditional Monte Carlo Variable X1(1:paths)=X_0;%Iterated Integral Monte Carlo Variable for nn=1:T_index %Loop over time. Random0=randn(size(X)); %Stochastic integrals follow dZn=Random0*sqrt(dt); dZndt(1:paths)=dt.^1.5/sqrt(3)*Random0; dtdZn(1:paths)=dt.^1.5/sqrt(3)*Random0; dZndZn(1:paths)=dt/2.0.*(Random0.^21); dtdt=dt.^2.0/2.0; X=X+mu0.*X*dt+sigma0*X.*dZn; %Discretization of traditional Monte Carlo X1=X1+mu0.*X1*dt+sigma0*X1.*dZn ... %%Discretization of Iterated Integral +mu0.*mu0.*X1.*dtdt ... %% Monte Carlo +mu0.*sigma0.*X1.*dZndt ... +sigma0.*mu0.*X1.*dtdZn ... +sigma0.*sigma0.*X1.*dZndZn; end %Following block is the analytic solution of lognormal SDE with drift. n=0; for n=1:2000 n=n+1 Z(n)=n*.005; Logf(n)=1.0/sqrt(2*pi)/Z(n)./(sigma0*sqrt(TT)).* ... exp(.5*(log(Z(n))log(X_0)(mu0.5*sigma0.^2)*TT).^2./(sigma0.^2*TT)); end %The next block turns monte carlo diffusion into density. MaxCutOff=50; BinSize=.01; [XDensity,IndexOutX,IndexMaxX] = MakeDensityFromSimulation_Infiniti(X,paths,BinSize,MaxCutOff ); [XDensity1,IndexOutX1,IndexMaxX1] = MakeDensityFromSimulation_Infiniti(X1,paths,BinSize,MaxCutOff ); plot(IndexOutX(1:IndexMaxX),XDensity(1:IndexMaxX),'r',IndexOutX1(1:IndexMaxX1),XDensity1(1:IndexMaxX1),'g', ... Z(1:2000),Logf(1:2000),'k') %In the graph red line is traditional monte carlo with Euler method. %green line is iterated integral monte carlo %black line is analytic solution str=input('look at overlaid Graph comparison of Monte carlo density of SDE using iterated integral(green line) with density of SDE generated from analytic method(black line)>');
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%5 Below is another function that is used for graphing the monte carlo diffusion and is called in the above program.
function [XDensity,IndexOut,IndexMax] = MakeDensityFromSimulation_Infiniti(X,Paths,BinSize,MaxCutOff ) %Processes monte carlo paths to return a series Xdensity as a function of IndexOut. IndexMax is the %maximum value of index. %
Xmin=0; Xmax=0; for p=1:Paths if(X(p)>MaxCutOff) X(p)=MaxCutOff; end if(Xmin>real(X(p))) Xmin=real(X(p)); end if(XmaxXmax=real(X(p)); end end
IndexMax=floor((XmaxXmin)/BinSize+.5)+1 XDensity(1:IndexMax)=0.0; for p=1:Paths index=real(floor(real(X(p)Xmin)/BinSize+.5)+1); if(real(index)<1) index=1; end if(real(index)>IndexMax) index=IndexMax; end XDensity(index)=XDensity(index)+1.0/Paths/BinSize; end IndexOut(1:IndexMax)=Xmin+(0:(IndexMax1))*BinSize; endAttached File: SVMonteCarloHigherOrderInfiniti.zip Attached File: MakeDensityFromSimulation_Infiniti.zip 

