My goal is to find the expected value of a sum of correlated standard normal random variables. Suppose that $f$ is some function and $p_1,p_2,...,p_n$ are polynomials of degree $m$. Furthermore, $X_1,X_2,...X_n$ are all standard normal with correlation coefficient $\rho$.
I want to find an efficient way to evaluate \begin{equation}\mathbb{E}\left[f\left(\sum p_i(X_i)\right)\right]\end{equation}
Obviously, this could be done by integrating over all $X_i$. But this is numerically to expensive. I was hoping to find a way to only use a single integral, given that the random variables are all equally distributed.
It would also be sufficient to find the characteristic function of $\sum p_i(X_i)$, which looks like it should be possible, but I failed so far.
"$f$ is some function" makes the problem impossible. With $f$ being anything (including integrable, I suppose), the resulting random variable $Z=f\left(\sum p_i(X_i)\right)$ can have any distribution, and you are asking for a method to calculate any expectation effectively (whatever this means).
That said, (and amazingly) Monte-Carlo will work!