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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 ?2011-08-11

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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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    @Simon Are they supposed to be bounded? What would be the forward Kolmogorov's equation, as written by Gortaur with $a=a(x,m)$ or something else?2019-02-19