How to approximate $\mathbb{E}\int_0^t f(s,X_s)ds$?

Question

I'am looking for any numerical methods to approximate $\mathbb{E}\int_0^t f(s,X_s)ds$, where $f:[0,t]\times \mathbb{R}^d \rightarrow\mathbb{R}$ is know and $X_s$ is a Ito process, we can assume that we know exact distribution of $X_s$ and we are able to to generate it. I will be grateful for any suggestions.

Edit: To be more precise: Is it any numerical method (even with additional assumptions about $f$ and $X$) to approximate $\mathbb{E}\int_0^t f(s,X_s)ds$ which reaches lower error in $L^2$ norm than method described by @Youem:

for fixing $0=_0<_1<\dots<_=$ (equidistant) and $$

For $=1,\dots,$, generate the process on $_$, $^{()}_{_0},\dots,^{()}_{_{−1}}$

$$\mathbb{E}\int_0^t f(s,X_s)ds\approx\frac{1}{M}\sum_{m=1}^M\sum_{k=0}^{n-1}(s_{i+1}-s_i)f(s_i,X_{s_i}^{(m)})$$

Answer 1

How about fixing $0 = s_0 < s_1 < \cdots < s_n = t$ and $M$

• For $m = 1,\ldots,M$, generate the process on $s_i$,

$$X^{(m)}_{s_0},\ldots,X^{(m)}_{s_{n-1}}$$

• Take:

$$\frac1M \sum_{m=1}^M \sum_{k=0}^{n-1} \left(s_{i+1} - s_i\right) f\left(s_i, X^{(m)}_{s_i}\right)$$

You can probably improve the approximation by par example taking the midpoints as evaluation points or any other integral approximation.