On Hamilton-Jacobi-Bellman equations with convex gradient constraints

TL;DR

This paper investigates a class of PDEs with convex gradient constraints arising from stochastic control problems, establishing conditions for unique solutions with Lipschitz continuous gradients and generalizing previous linear cases.

Contribution

It identifies conditions for uniqueness and regularity of solutions to Hamilton-Jacobi-Bellman equations with convex gradient constraints, extending prior linear maximum cases.

Findings

Established uniqueness of solutions under certain conditions.

Proved Lipschitz continuity of solutions' gradients.

Generalized previous results to broader convex cases.

Abstract

We study PDE of the form $\max\{F(D^2u,x)-f(x), H(Du)\}=0$ where $F$ is uniformly elliptic and convex in its first argument, $H$ is convex, $f$ is a given function and $u$ is the unknown. These equations are derived from dynamic programming in a wide class of stochastic singular control problems. In particular, examples of these equations arise in mathematical finance models involving transaction costs, in queuing theory, and spacecraft control problems. The main aspects of this work are to identify conditions under which solutions are uniquely defined and have Lipschitz continuous gradients. We also generalize previous results known for the case where $M\mapsto F(M,x)$ is the maximum of finitely many linear functions.