# HJB equations with gradient constraint associated with controlled jump-diffusion processes

## TL;DR

This paper proves the existence and uniqueness of solutions to a complex Hamilton-Jacobi-Bellman equation with gradient constraints, arising in singular control problems involving jump-diffusion processes with finite variation jumps.

## Contribution

It establishes the well-posedness of a novel class of HJB equations with gradient constraints and partial integro-differential operators linked to controlled jump-diffusion models.

## Key findings

- Existence and uniqueness of solutions to the HJB equation.
- Verification that the value function satisfies the HJB equation.
- Application to singular control problems with jump processes.

## Abstract

In this paper, we guarantee the existence and uniqueness (in the almost everywhere sense) of the solution to a Hamilton-Jacobi-Bellman (HJB) equation with gradient constraint and a partial integro-differential operator whose L\'evy measure has bounded variation. This type of equation arises in a singular control problem, where the state process is a multidimensional jump-diffusion with jumps of finite variation and infinite activity. We verify, by means of {\epsilon}-penalized controls, that the value function associated with this problem satisfies the aforementioned HJB equation.