I am analyzing the following problem: given a set of players $x^i_t$ for $i=1,\dots,N$ satisfying the SDE $$ dx^i_t = \alpha^i_t dt + \sigma dW^i_t $$ where $W^i_t$ are independent Brownian Motions, and $\alpha^i_t$ is the control function, solution of the following optimization problem $$ J^i(\alpha^i_\cdot) = \mathbb{E}\left[ \int_0^T F^i(\mathbf{x}_t)\,dt\right] $$ where $F^i$ is given (with suitable hypothesis) and $\mathbf{x}_t = (x^1_t,\dots, x^N_t)$. In this set up player $i$ wants to maximize it's functional $J^i$. Here the function $F^i$ is intended to be of mean field type, i.e. is a functional on measure spaces, depending on the empirical measure of the system $$ \mu^N_t = \frac{1}{N}\sum_{i=1}^{N} \delta_{x^i_t}. $$
My question is: is it possible to solve this kind of problem numerically, meaning producing samples of $x^i_t, \alpha^i_t$, for a finite number of players?
If yes is there a canonical reference on the topic? By looking online I found many references that solve the problem in the limit $N \to \infty$ but never speaks about the finite player case.