Extended mean field control: a global numerical solution via finite-dimensional approximation

We investigate the global numerical approximation of a class of extended mean field control problems (MFC), where the dynamics and costs depend on the joint distribution of the state and the control. We propose a framework to approximate the value function globally over the Wasserstein space, moving beyond the restriction of fixed initial conditions. Our approach exploits the propagation of chaos by approximating the infinite-dimensional MFC problem by an $N$-player cooperative game, together with the usage of finite-dimensional solvers. This method avoids the need to parametrise functions on an infinite-dimensional space, offering a balance between probabilistic rigor and computational efficiency.

💡 Research Summary

The paper tackles the challenging problem of globally approximating the value function of extended mean‑field control (MFC) problems, where both the dynamics and the cost depend on the joint distribution of the state and the control. Traditional approaches either restrict themselves to a fixed initial distribution or rely on particle simulations that only provide solutions for particular initial conditions. The authors propose a novel framework that approximates the intrinsic value function (v(t,\mu)) over the whole Wasserstein space without discretising the space of probability measures.

The key methodological insight is to exploit the propagation of chaos: the infinite‑dimensional MFC problem is approximated by a cooperative N‑player game whose state dynamics are standard McKean‑Vlasov SDEs with i.i.d. initial distribution (\mu). The value function of this finite‑player game, denoted (\bar v_N(t,x_1,\dots,x_N)), lives in a finite‑dimensional Euclidean space. Under standard Lipschitz and boundedness assumptions on the drift, diffusion, running cost and terminal cost, the authors prove that as (N\to\infty) the empirical distribution of the N particles yields a consistent estimator of the original value function: \