Finite Dimensional Approximations of Hamilton–Jacobi–Bellman Equations for Stochastic Particle Systems with Common Noise

.This paper is a continuation of the program started in [W. Gangbo, S. Mayorga, and A. Święch, SIAM J. Math. Anal., 53 (2021), pp. 1320–1356]. We consider stochastic optimal control problems for systems of \(n\) particles with common noise, interacting through their empirical measures. We prove that their value functions \(u_n\) , when converted to functions of the empirical measures, converge uniformly on bounded sets in the Wasserstein space of probability measures to a limiting function \(V\) which is the unique so-called \(L\) -viscosity solution of the limiting Hamilton–Jacobi–Bellman (HJB) equation in the Wasserstein space. This generalizes such a result from the 2021 paper, in the sense that our drift and cost functions are allowed to be unbounded. The \(L\) -viscosity solution is defined using the lifting of the function and the HJB equation to a Hilbert space of square integrable random variables. The proof of the convergence of \(u_n\) to \(V\) uses PDE viscosity solution techniques. The lifted function \(V\) of \(V\) is the value function of a stochastic optimal control problem in the Hilbert space and we prove that \(V\) is \(C1,1\) and that the problem has an optimal feedback control. Using this we then characterize \(V\) as the value function of a stochastic optimal control problem in the Wasserstein space.