Analysis of Hamilton-Jacobi-Bellman equations arising in stochastic singular control

TL;DR

This paper investigates a class of Hamilton-Jacobi-Bellman equations related to stochastic singular control, establishing existence, uniqueness, and regularity properties of viscosity solutions under certain convexity conditions.

Contribution

It provides new results on the existence, uniqueness, and regularity of solutions to a key PDE model in stochastic singular control, especially under uniform convexity of H.

Findings

Existence of a unique viscosity solution with Hölder continuous gradient.

Lipschitz continuity of the gradient when H is uniformly convex.

The PDE model effectively describes stochastic singular control problems.

Abstract

We study the partial differential equation max{Lu - f, H(Du)}=0 where u is the unknown function, L is a second-order elliptic operator, f is a given smooth function and H is a convex function. This is a model equation for Hamilton-Jacobi-Bellman equations arising in stochastic singular control. We establish the existence of a unique viscosity solution of the Dirichlet problem that has a Holder continuous gradient. We also show that if H is uniformly convex, the gradient of this solution is Lipschitz continuous.