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Hi there,

Suppose I have a diffusion process

$dX_t = a(X_t)dt + b(X_t)dW_t$. Is there a straightforward method for approximating the first few moments of $X_T$ for some time $T$? Clearly, one could use Monte-Carlo methods, but I'd like something a bit more analytical.

Is it possible to use stochastic Taylor expansions to find such an estimate, for example?

Many thanks.

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How about first computing the solution to $dX_t = a(X_t) dt$ and then linearizing the system about this. I think you could get a fairly explicit expression for the moments then. This approach would work if $b$ is small. On the other hand, if $T$ is small then the stochastic Taylor expansion would work. – Paul Tupper Aug 12 '11 at 13:50
The only reference I've seen for such a linearization is in Gardiner's book, in the section on small noise expansions. These are sometimes highly unstable. For instance, when $a(x) = x(1-x^2)$, the linearization proposed by Gardiner actually grows exponentially quickly. If you've got an alternative reference, I'd love to hear about it. – Simon Lyons Aug 12 '11 at 14:32
Yeah, in that case the solution for reasonable time lengths is not going to be close to the solution with $b=0$. Two thoughts: 1. Since you are only interested in moments at $T$, everything you want is going to be in the solution of the Fokker-Planck equation. The stochastic taylor expansion is not going to help you. 2. If $b$ is small but $T$ is big, then I think this is the territory covered by Freidlin and Wentzell's Random Perturbations of Dynamical Systems. – Paul Tupper Aug 13 '11 at 21:42

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