Solution to HJB equations with an elliptic integro-differential operator and gradient constraint

TL;DR

This paper proves existence, regularity, and uniqueness for solutions to a Hamilton-Jacobi-Bellman equation involving an elliptic integro-differential operator, relevant in stochastic control with Levy processes.

Contribution

It establishes foundational results for HJB equations with elliptic integro-differential operators in the context of Levy process control problems.

Findings

Existence of solutions proven under certain conditions

Regularity results for the solutions obtained

Uniqueness of solutions established

Abstract

The main goal of this paper is to establish existence, regularity and uniqueness results for the solution of a Hamilton-Jacobi-Bellman (HJB) equation, whose operator is an elliptic integro-differential operator. The HJB equation studied in this work arises in singular stochastic control problems where the state process is a controlled $d$-dimensional L\'evy process.