Now, I have a stochastic differential equation, $dx=f(x)dt+h(x)dB$ with intitial condition $x_0$, and a random varaible $g\triangleq \exp\{\int_0^{t} F(x_s) ds\} $. Here, $f(\cdot)$, $h(\cdot)$ and $F(\cdot)$ are arbitrary functions. My question is whether there is some existing formula to calculate the expectation of g (that is $\mathbb{E}(g)$).
I am not a native English speaker. Please let me know if my question is vague. I will correct it immediately if you point out. Also, I will really appreciate if you could answer this question or refer some literature to me.