Sensitivity analysis for HJB equations with an application to coupled backward-forward systems

TL;DR

This paper investigates how solutions to Hamilton-Jacobi-Bellman equations depend on parameters, providing Lipschitz continuity results that facilitate solving coupled backward-forward systems in mean field games.

Contribution

It establishes Lipschitz continuous dependence of HJB solutions on parameters and applies these results to coupled backward-forward equations in mean field games.

Findings

Solutions are Lipschitz continuous with respect to parameters.

Verifiable criteria for feedback regularity are provided.

Application to solving coupled backward-forward systems.

Abstract

In this paper, we analyse Lipschitz continuous dependence of the solution to Hamilton-Jacobi-Bellman equations on a functional parameter. This sensitivity analysis not only has the interest on its own, but also is important for the mean field games methodology, namely for solving a coupled system of backward-forward equations. We show that the unique solution to a Hamilton-Jacobi-Bellman equation and its spacial gradient are Lipschitz continuous uniformly with respect to the functional parameter. In particular, we provide verifiable criteria for the so-called feedback regularity condition. Finally as an application, we show how the sensitive results are used to solved the coupled system of backward-forward equations.