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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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