Sorry It was too late yesterday when I wrote the above post and it took me more than one and a half hour to write it and I slept and woke up several times on my desk in between. So I resume yesterday's post.Dear friends, I was trying to think how to derive the proper equations for the evolution of density in Ito-hermite method. I was able to find some other interesting equations that I decided to share here with friends.

Before I write the main equations, I want to mention that this idea is based on equivalence between two densities. I take one density that is standard normal called Z (could also be taken as brownian motion) and then I try to find the evolution of a particular point on the density of SDE variable X(t,Z) and this particular point is related to a specific fixed value of Z which is already known. Z is a static density that does not change with time at all.

I will write a few equations first that we will use later in the main derivation. Here are the equations

[$]P(Z)=P(X,t) |{\frac{\partial X}{\partial Z}}|[$] Follows from change of variable formula for densities.

[$]\frac{\partial P(X,t)}{\partial X}= \frac{\partial p(Z)}{\partial Z} (\frac{\partial Z}{\partial X})^2[$]

[$]\frac{d}{dt} \big[ {\frac{\partial X}{\partial Z}} \big]=\frac{\partial}{\partial X}\big[ {\frac{\partial X}{\partial Z}} \big] \frac{\partial X}{\partial t}[$]

[$]=-{(\frac{\partial X}{\partial Z})}^2 \frac{\partial^2 Z}{\partial X^2} \frac{\partial X}{\partial t}[$]

[$]\frac{\partial P(X,t)}{\partial t} = f(X)[$] where f(X) is the fokker planck equation.

Now we come towards the main equation derivation(which might as well possibly be wrong)

Since standard normal is a static density and it does not change with time, we can write

[$]\frac{d}{dt}[p(Z)]=0[$]

since we can write densities in terms of each other using change of variable formula, we can write

[$]\frac{d}{dt}[p(X,t)\frac{\partial X}{\partial Z}]=0[$]

taking the derivatives, we can expand the above equation as

[$]\big [ \frac{\partial p(X,t)}{\partial t}+\frac{\partial p(X,t)}{\partial X}\frac{\partial X}{\partial t} ]\frac{\partial X}{\partial Z} [$]

[$]-p(X,t){(\frac{\partial X}{\partial Z})}^2 \frac{\partial^2 Z}{\partial X^2} \frac{\partial X}{\partial t}=0[$]

using

[$]P(Z)=P(X,t) |{\frac{\partial X}{\partial Z}}|[$]

[$]\frac{\partial P(X,t)}{\partial X}= \frac{\partial p(Z)}{\partial Z} (\frac{\partial Z}{\partial X})^2[$]

and [$]\frac{\partial P(X,t)}{\partial t} = f(X)[$]

in the above equation ,we get

[$]\frac{\partial X(Z,t)}{\partial t} =\frac{ \frac{\partial p(X,t)}{\partial t}}{ \big[ (p(Z) \frac{\partial^2 Z}{\partial X^2}+ Z p(Z) ({\frac{\partial Z}{\partial X}})^2\big]}[$]

using the above equation, we can probably solve every SDE so that we can solve for evolution of X as a function of Z and t. More detailed post and notes tomorrow.

copying from the above post we have

[$]\frac{\partial X(Z,t)}{\partial t} =\frac{ \frac{\partial p(X,t)}{\partial t}}{ \big[ (p(Z) \frac{\partial^2 Z}{\partial X^2}+ Z p(Z) ({\frac{\partial Z}{\partial X}})^2\big]}[$]

where [$]\frac{\partial p(X,t)}{\partial t}[$] is the fokker planck equation for the related SDE.

We can use the change of variable formula in fokker-planck again to get rid of the densities and get an evolution equation solely in terms of Z and SDE variable X(Z,t). Suppose the structure of fokker planck is given as

[$]\frac{\partial p(X,t)}{\partial t}=a(X,t) + b(X,t) p(X,t) + c(X,t) \frac{\partial p(X,t)}{\partial X} + d(X,t) \frac{\partial^2 p(X,t)}{\partial X^2}[$]

when we have [$]a(X,t)=0[$] we can eliminate the densities from the equation by converting p(X,t) into p(Z) and cancelling the p(Z) in the equation. using the following formulas, we get

[$]p(X)=p(Z) |\frac{\partial Z}{\partial X}|[$] This follows from the standard change of variables in a density.

[$]\frac{\partial p(X)}{\partial X}=\frac{\partial p(Z)}{\partial Z} {(\frac{\partial Z}{\partial X})}^2+p(Z) \frac{\partial^2 Z}{\partial X^2} =-Z p(Z){(\frac{\partial Z}{\partial X})}^2+p(Z) \frac{\partial^2 Z}{\partial X^2} [$]

[$]\frac{\partial^2 p(X)}{\partial X^2}=\frac{\partial^2 p(Z)}{\partial Z^2} {(\frac{\partial Z}{\partial X})}^3+3 \frac{\partial p(Z)}{\partial Z} \frac{\partial Z}{\partial X} \frac{\partial^2 Z}{\partial X^2}+p(Z) \frac{\partial^3 Z}{\partial X^3}[$]

[$]=(Z^2-1) p(Z) {(\frac{\partial Z}{\partial X})}^3 -3 Z p(Z) \frac{\partial Z}{\partial X} \frac{\partial^2 Z}{\partial X^2} +p(Z) \frac{\partial^3 Z}{\partial X^3}[$]

[$]\frac{\partial p(X,t)}{\partial t}= b(X,t) p(Z) |\frac{\partial Z}{\partial X}| + c(X,t) \big[-Z p(Z){(\frac{\partial Z}{\partial X})}^2+p(Z) \frac{\partial^2 Z}{\partial X^2}\big][$]

[$]+ d(X,t) \big[ (Z^2-1) p(Z) {(\frac{\partial Z}{\partial X})}^3 -3 Z p(Z) \frac{\partial Z}{\partial X} \frac{\partial^2 Z}{\partial X^2} +p(Z) \frac{\partial^3 Z}{\partial X^3} \big][$]

putting the above value of fokker planck in the first equation and cancelling p(Z) we get

[$]\frac{\partial X(Z,t)}{\partial t} =\frac{ b(X,t) |\frac{\partial Z}{\partial X}| + c(X,t) \big[-Z {(\frac{\partial Z}{\partial X})}^2+ \frac{\partial^2 Z}{\partial X^2}\big] + d(X,t) \big[ (Z^2-1) {(\frac{\partial Z}{\partial X})}^3 -3 Z \frac{\partial Z}{\partial X} \frac{\partial^2 Z}{\partial X^2} + \frac{\partial^3 Z}{\partial X^3} \big]}{ \big[ \frac{\partial^2 Z}{\partial X^2}+ Z ({\frac{\partial Z}{\partial X}})^2\big]}[$]

Please be careful with signs as I might have missed some absolute in change of probability derivatives.

In order to work out a proper evolution equation, We can take the value of Z and its derivatives [$]\frac{\partial Z}{\partial X}[$] and [$]\frac{\partial^2 Z}{\partial X^2}[$] and [$]\frac{\partial Z^3}{\partial X^3}[$] as constant at start of the time step and expand the integrals of the kind like [$]\int_0^t b(X,s) H_n(Z) ds[$], [$]\int_0^t c(X,s)H_n(Z) ds[$] and [$]\int_0^t d(X,s) H_n(Z) ds[$] for a particular value of Z related to each grid point using stochastics just like we did for monte carlo simulations and then form a valid evolution equation. In the above integrals [$]H_n(Z)[$] is a standard normal hermite polynomial of nth degree.

These are very basic thoughts that I just worked out quickly and there is also some possibility that we might have to use the equivalence relation with brownian motion density instead of standard normal density but if required doing the same exercise on brownian motion should also be straightforward.

I will be doing my experiments and coming out with a more detailed post in a day.

Again since I recently did this and never numerically tried it there is a chance that we might have to use brownian motion density instead of standard normal density but that should also be straightforward.