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

Abstract

.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 \(\mathcal{V}\) 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 \(\mathcal{V}\) uses PDE viscosity solution techniques. The lifted function \(V\) of \(\mathcal{V}\) is the value function of a stochastic optimal control problem in the Hilbert space and we prove that \(V\) is \(C^{1,1}\) and that the problem has an optimal feedback control. Using this we then characterize \(\mathcal{V}\) as the value function of a stochastic optimal control problem in the Wasserstein space.KeywordsHamilton–Jacobi–Bellman equations in infinite dimensionviscosity solutionsmean field controlstochastic particle systemsmaster equationWasserstein spaceWasserstein LaplacianMSC codes35D4035F2135R1549L2549N8093E20