I have the following two OU processesandwhere W_1 and W_2 are independent Brownian motions.LetI am interested in the following expectationI derived the dynamics of f but (of course) the drift is a non-linear function of X_1 and X_2 so I can not use standard affine arguments to get a closed form expression. I guess I can calculate the expectation using finite difference methods, but because I have to evaluate this expectation many times I was wondering if there is some other solution. Any ideas?Thanks in advance!

Maybe look for an expansion justified by small or large (dimensionless) parameters. For example, try E[f(T) exp{-int(0,T) f(s) ds}] = E[f(T) exp(-Z)] = E[f(T)(1 - Z +Z^2/2 + ...)] and see how many terms you can evaluate in E[...]; Then compare with a numerical pde soln to see if it's any good in your parameter ranges.This is perhaps a "small sigma^2 T, small kappa T" expansion.Or expand exp(-Z) about exp(-E[Z]), or maybe the large T asymptotics are good enough, etc.Unfortunately, this kind of thing hardly ever works if you are trying to "optimize" parameters.The reason being that the optimizer will probably explore regions of the parameters space where anygiven approximation breaks down badly. If that's what you're doing, you're faced with a *very* difficultcomputation. It would be difficult even with one OU factor. My advice would be to try a more tractable modelthat has the essential features you need.

Thank you for the suggestions!It seems a quadratic specification would indeed help.