Finite Dimensional Approximations of Hamilton--Jacobi--Bellman Equations in Spaces of Probability Measures

Abstract

We prove that viscosity solutions of Hamilton--Jacobi--Bellman (HJB) equations, corresponding either to deterministic optimal control problems for systems of $n$ particles or to stochastic optimal control problems for systems of $n$ particles with a common noise, converge locally uniformly to the viscosity solution of a limiting HJB equation in the space of probability measures. We prove uniform continuity estimates for viscosity solutions of the approximating problems which may be of independent interest. We pay special attention to the case when the Hamiltonian is convex in the gradient variable and equations are of first order and provide a representation formula for the solution of the limiting first order HJB equation. We also propose an intrinsic definition of viscosity solution on the Wasserstein space.