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Suppose I have a diffusion $dX_t = a(X_t)dt + b(X_t)dW_t$. Is there a straightforward way of estimating the variance of $X_T$ for some time $T$, assuming that $T$ is large enough so that a simple Euler approximation isn't accurate?

Clearly, Monte-Carlo methods could be used here, but I'd like something more analytical.

Any ideas?

Many thanks.

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What conditions over a and b ? are they deterministic, are they adapted ? how smooth are they ? –  TheBridge Aug 11 '11 at 22:39

1 Answer 1

One of the methods is the forward Kolmogorov's equation: $$ \begin{cases} \frac{\partial m}{\partial t} &= a(x)\frac{\partial m}{\partial x} + b(x)\frac{\partial^2 m}{\partial x^2}, \\ m(0,x) &= f(x) \end{cases} $$ where $m(t,x) = \mathsf E_xf(X_t) = \mathsf E[f(X_t)|X_0=x]$.

In your case you should make calculations for $f_1 = x$ and $f_2 = x^2$. Then the variance will be given by $$ V[X_T] = m_2(T,x) - (m_1(T,x))^2 $$

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Does $\mathsf E_xf(X_t)$ stand for expectation with respect to distribution of $X_t$ with initial condition being $x$ ? –  Sasha Aug 11 '11 at 15:42
    
@Sasha: Yes, it does. Thank you for the comment, I will put it in the answer. –  Ilya Aug 11 '11 at 15:49
    
Thanks for your response. I'll most likely end up working in high dimensions, so the FKE isn't really an option. I'm looking for something a good deal cheaper, computationally speaking. On the other hand, it doesn't need to be as accurate either. A rough estimate would do. –  Simon Aug 11 '11 at 16:07
    
@Simon: as far as I know, there are MC methods and FKE. I also would be happy to learn if there are others. The usual point is tha MC are working better in large dimensions. If you're looking for the steady behavior ($T\approx \infty$) or $T$ is just a large number (sorry for the informal statement)? –  Ilya Aug 11 '11 at 16:20
    
I'm just assuming $T$ is a large-ish number. A steady state does not necessarily exist. I was wondering if I could get some useful information from stochastic Taylor expansions, maybe. –  Simon Aug 11 '11 at 16:37

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