In my previous post, I suggested to expand the SDE variable [$]X(t)[$] as a function of hermite polynomials of the brownain motion but then I realized that drift and other functions of X(t) are basically non-linear functions and they non-linearly affect various hermite polynomials in the hermite polynomial expansion of X(t). And then I thought that if we can expand the drift and other functions of X(t) in terms of hermite polynomials, we would probably not even need to solve the fokker planck equation at all. Here are some of my more developed thoughts that I want to share with friends but friends need to think analytically on their own since I have not tested my ideas with experiments and there might be mistakes in untested ideas. But I hope that there is enough food for thought to share the ideas with friends at this stage.

The main theme here is to find an analytical expansion of X(t) in terms of hermite polynomials and then find hermite expressions of [$]\mu(X)[$] and [$]\sigma(X)[$] and then analytically evolve the coefficients of hermite expansion of [$]X(t)[$] over each time step

First some basics, we are dealing with hermite polynomials in standard normal [$]Z[$] but most of it is directly applicable to hermite polynomials of broawnian motion [$]B(t)[$] as well.

Differentiation of hermite polynomials [$]\frac{d H_{n+1}(Z)}{dZ}=(n+1) H_{n}(Z)[$]

repeated integrations of integral over normal random variable as

[$]\int_0^Z dZ_1=Z=H_1(Z)[$]

[$]\int_0^Z \int_0^{Z_1} dZ_2 dZ_1=\frac{Z^2-1}{2}=\frac{1}{2} H_2(Z)[$]

[$]\int_0^Z \int_0^{Z_1} \int_0^{Z_2} dZ_3 dZ_2 dZ_1=\frac{1}{6} (Z^3-3Z)=\frac{1}{6}H_3(Z)[$]

and so on.

First of all we will deal with the case of Lamperti form of SDEs since it is lot easier to explain with very little latexing and lesser equations.

Suppose SDE under consideration is [$]X(t)=\mu(X) dt + \sigma dB(t)[$]

When we start the evolution of a density from delta source, the very initial small time step form is essentially a normal. We can denote the first step evolution as a Hermite form as given by equation [$]X(Z)=X_0+\mu(X_0) \Delta t + \sigma \sqrt{\Delta t} Z=X_0+\mu(X_0) \Delta t + \sigma \sqrt{\Delta t} H_1(Z)[$]

We can write it as [$]Z=a_0+a_1 H_1(Z)[$] where [$]a_0=X_0+\mu(X_0) \Delta t[$] and [$]a_1=\sigma \sqrt{\Delta t}[$]

Now when we take the second step in the evolution of SDE, it is more involved since we are not starting from a delta origin anymore. Obviously we have to add the volatility in a squared fashion to second hermite polynomial but we also have to represent the drift as a function of hermite polynomials and add it linearly to the existing one step evolved hermite representation of X(t).

Next we have to learn how to expand functions of X(t) in terms of hermite polynomials once we are given a hermite representation of X(t) itself. We take [$]\mu(X)[$] as a test function and expand it in terms of hermite polynomials. This has to follow iterated integrals form of Taylor series as

[$]\mu(X(Z))=\mu(X(Z_0)) + \int_0^Z \frac{d \mu(X)}{dX} \frac{dX}{dZ} dZ[$]

Above we have expanded the [$]\mu(X)[$] to one term and evaluating the integral at [$]X_0=X(Z_0)[$] and expanding one step further, we get

[$]\mu(X(Z))=\mu(X(Z_0)) + \frac{d \mu(X_0)}{dX} \frac{dX_0}{dZ} \int_0^Z dZ_1 [$]

[$]+ \int_0^Z \int_0^{Z_1} \big[ \frac{d^2 \mu(X)}{dX^2} {(\frac{dX}{dZ})}^2 +\frac{d \mu(X)}{dX} \frac{dX^2}{dZ^2} \big] dZ_2 dZ_1 [$]

When expanded to thirs hermite polynomial, drift can be written as

[$]\mu(X(Z))=\mu(X(Z_0)) + \frac{d \mu(X_0)}{dX} \frac{dX_0}{dZ} \int_0^Z dZ_1 [$]

[$]+\big[ \frac{d^2 \mu(X_0)}{dX^2} {(\frac{dX}{dZ})}^2 +\frac{d \mu(X_0)}{dX} \frac{dX^2}{dZ^2} \big] \int_0^Z \int_0^{Z_1} dZ_2 dZ_1 [$]

[$]+\big[ \frac{d^3 \mu(X_0)}{dX^3} {(\frac{dX}{dZ})}^3+3 \frac{d^2 \mu(X_0)}{dX^2} {(\frac{dX}{dZ})}^2 \frac{d^2 X}{dZ^2}+\frac{d \mu(X_0)}{dX} \frac{dX^3}{dZ^3} \big] \int_0^Z \int_0^{Z_1} \int_0^{Z_2} dZ_3 dZ_2 dZ_1 + ..... [$]

which will be equivalent when written in form of hermite polynomials as

[$]\mu(X(Z))=\mu(X(Z_0)) + \frac{d \mu(X_0)}{dX} \frac{dX_0}{dZ} H_1(Z) [$]

[$]+\big[ \frac{d^2 \mu(X_0)}{dX^2} {(\frac{dX}{dZ})}^2 +\frac{d \mu(X_0)}{dX} \frac{dX^2}{dZ^2} \big] \frac{H_2(Z)}{2} [$]

[$]+\big[ \frac{d^3 \mu(X_0)}{dX^3} {(\frac{dX}{dZ})}^3+3 \frac{d^2 \mu(X_0)}{dX^2} {(\frac{dX}{dZ})}^2 \frac{d^2 X}{dZ^2}+\frac{d \mu(X_0)}{dX} \frac{dX^3}{dZ^3} \big] \frac{H_3(Z)}{6} + O (H_4(Z))[$]

All derivatives in the above are taken at [$]X=X(Z_0)[$]. We have to take enough terms in hermite expansion of functions of X(t) so as to be able to faithfully follow the true function over the range of Z=-4 to +4. In the above equation we can see that even though the hermite representation of X(t) can take just one derivative, the hermite representation of functions of X(t) could be far more complex and could take significantly many more hermite polynomials in expansion.

Continuing after one step evolution of Lamperti SDE given as [$]X(Z)=a_0+a_1 H_1(Z)[$], we get a representation for drift as

[$]\mu(X(Z))=\mu(X(Z_0)) + \frac{d \mu(X_0)}{dX} a_1 H_1(Z) [$]

[$]+\frac{d^2 \mu(X_0)}{dX^2} {a_1}^2 \frac{H_2(Z)}{2}+\frac{d^3 \mu(X_0)}{dX^3} {a_1}^3 \frac{H_3(Z)}{6}[$]

The above expression for drift could be linearly added by each hermite polynomial to the first one step evolution of X(t) but first, in case of Lamperti, we would have to add the one step volatility in a squared fashion to first order hermite polynomial and then linearly add the drift and then move on step by step for new updated representations of drift and adding them to the previous stage hermite representation of X(t) and also adding volatility in a squared fashion to first hermite polynomial. Particularly after two steps from the delta origin, the hermite representation of X(t) would be

[$]X(Z)=X_0+ \mu(X_0) \Delta t + \mu(X(Z_0)) \Delta t + [ \sigma \sqrt{\Delta t +\Delta t} + \frac{d \mu(X_0)}{dX} \Delta t \sigma \sqrt{\Delta t}] H_1(Z) [$]

[$]+\frac{d^2 \mu(X_0)}{dX^2} {( \sigma \sqrt{\Delta t})}^2 \Delta t \frac{H_2(Z)}{2}+\frac{d^3 \mu(X_0)}{dX^3} {( \sigma \sqrt{\Delta t})}^3 \Delta t \frac{H_3(Z)}{6}[$]

where I have used [$]a_1=\sigma \sqrt{\Delta t}[$]

The third step would be even more complicated since third step would get a contribution in expansion of [$]\mu(X)[$] in terms of hermite polynomials from large number of X(t) expansion hermite terms from step two.

Now we come towards a general evolution algorithm using hermite polynomials. First we take a one step evolution equation in terms of [$]\mu(X)[$], [$]\sigma(X)[$], and their derivatives with respect to X(t) and hermite polynomials of brownian motion[$]B(t)[$]. Secondly we find hermite representation of all functions of drift and volatility and their derivatives in terms of X(t) in the above terms in terms of hermites of normal random variable around [$]Z=0[$]. And then we add all volatility terms to existing previous step hermite representation of X(t) in a squared fashion and then we add all drift terms in hermite form to the hermite form of X(t) in a linear fashion.

Lamperti form is very attractive since it would take far smaller no of hermite polynomials in its expansion as its density remains reasonably similar to normal random variable density and we do not need to expand the volatility in this case.

I will soon be distributing the worked out code with this evolution scheme and then we move to full fledged solution of Fokker-Planck equation.