On Inverse Problems for Mean Field Games with Common Noise via Carleman Estimate

TL;DR

This paper investigates inverse problems in mean field games with common noise, establishing stability and uniqueness results using new Carleman estimates for coupled stochastic PDE systems.

Contribution

It introduces novel Carleman estimates to prove stability and uniqueness in inverse problems for MFGs with common noise, a significant advancement in the field.

Findings

Established Lipschitz and H"older stability for the inverse problem.

Proved uniqueness for the inverse source problem.

Developed new Carleman estimates for stochastic PDE systems.

Abstract

In this paper, we study two kinds of inverse problems for Mean Field Games (MFGs) with common noise. Our focus is on MFGs described by a coupled system of stochastic Hamilton-Jacobi-Bellman and Fokker-Planck equations. Firstly, we establish the Lipschitz and H\"older stability for determining the solutions of a coupled system of stochastic Hamilton-Jacobi-Bellman and Fokker-Planck equations based on terminal observation of the density function. Secondly, we derive a uniqueness theorem for an inverse source problem related to the system under consideration. The main tools to establish those results are two new Carleman estimates.