Non-convex Hamilton-Jacobi equations with gradient constraints

TL;DR

This paper investigates non-convex Hamilton-Jacobi equations with gradient constraints, establishing new regularity results and exploring applications in risk theory and stochastic control.

Contribution

It provides the first optimal regularity results for solutions to these non-convex equations with gradient bounds, advancing understanding of their mathematical properties.

Findings

Established new regularity bounds for solutions

Identified the impact of gradient bounds on solution smoothness

Applied results to risk theory and stochastic control models

Abstract

We study non-convex Hamilton-Jacobi equations in the presence of gradient constraints and produce new, optimal, regularity results for the solutions. A distinctive feature of those equations regards the existence of a lower bound to the norm of the gradient; it competes with the elliptic operator governing the problem, affecting the regularity of the solutions. This class of models relates to various important questions and finds applications in several areas; of particular interest is the modeling of optimal dividends problems for multiple insurance companies in risk theory and singular stochastic control in reversible investment models.