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.
That's what the "mean field" part means, using LLN to aggregate the game in a way that no player has influence over the global dynamics of what is happening. This is closer to the old style of general equilibrium economics, where each agent is measure zero, and they only interact through aggregates like prices.
You sound like you are more interested in "differential games":
https://en.wikipedia.org/wiki/Differential_game
An example would be:
https://en.wikipedia.org/wiki/Princess_and_Monster_game
This is a zero sum game in which a monster is trying to catch a princess, so the game is about evasion. Each player has constraints on their ability to maneuver, and the question is whether the initial conditions, constraints, and boundary values allow the monster to catch the princess or not.
If you just like Brownian motions, as so many people do, they are popular in "Continuous Time Moral Hazard" problems. The profits of, say, a firm are driven by the effort of a manager, who is controlling the (usually geometric) Brownian motion. The owner of the firm provides incentives for the manager, who cannot observe effort but only the resulting profits which are a noisy measure of effort, so he can only try to incent the manager to behave the way the owner prefers based on available information.
Finally, there is alot of classical repeated games work in continuous time. In particular, there's the question of whether, once the game is repeated, players can maintain Pareto optimal outcomes by using a system of punishments for deviations from the preferred outcome; so, a ''continuous time folk theorem''. Skrypacz from Stanford and David Rahman from Minnesota have some interesting papers on whether this is true or not, for different kinds of monitoring structures, among other people.
If you google the phrases in quotes, you'll get references to the literature.