# Simulating conditional expectations

There is a multidimensional process X defined via its SDE (we can assume that its a diffusion type process), and lets define another process by $g_t = E[G(X_T)|X_t]$ for $t\leq T$.

I would like to simulate process $g_t$, i.e. discretize to use in a Monte-Carlo simulation. What is the best way to do it?

The two approaches I can think of is (i) use Feynman-Kac and Finite Differeces to get $g_t$ as a function of $X$ and $t$, simulate $X_t$ and calculate $g_t$ (ii) use some form of Longstaff-Schwarz algorithm

Is there any better/simpler method?

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@Grzenio : Why Longstaff-Schwartz should be used here this is not an optimal stopping problem ? –  The Bridge Dec 21 '11 at 7:41
@TheBridge, it seems that you can use the ideas from Longstaff-Schwartz to calculate the conditional expectation at any time t<T in the simulation. So the algorithm would be certainly simpler than the original for American options, but we would still need the regression bit. As I said, I would be grateful if someone suggests a simpler method. –  Grzenio Dec 21 '11 at 13:53
@Grzenio : As I said as you havn't shown an optimal stopping problem it is unclear why LS algorithm (which is quite simple to implement by the way) would be needed. Regards –  The Bridge Dec 21 '11 at 14:22