Regularity Properties of Viscosity Solutions of Integro-Partial Differential Equations of Hamilton-Jacobi-Bellman Type

TL;DR

This paper investigates the regularity of viscosity solutions to Hamilton-Jacobi-Bellman integro-partial differential equations, showing they are jointly Lipschitz and semiconcave under certain conditions, using stochastic analysis techniques.

Contribution

It establishes regularity properties of viscosity solutions for a class of integro-PDEs of HJB type, employing innovative stochastic transformations.

Findings

Viscosity solutions are jointly Lipschitz continuous.

Solutions are jointly semiconcave in space and time.

Results hold for all compact time intervals excluding the terminal time.

Abstract

We study the regularity properties of integro-partial differential equations of Hamilton-Jocobi-Bellman type with terminal condition, which can be interpreted through a stochastic control system, composed of a forward and a backward stochastic differential equation, both driven by a Brownian motion and a compensated Poisson random measure. More precisely, we prove that, under appropriate assumptions, the viscosity solution of such equations is jointly Lipschitz and jointly semiconcave in $(t,x)\in\Delta\times\R^d$, for all compact time intervals $\Delta$ excluding the terminal time. Our approach is based on the time change for the Brownian motion and on Kulik's transformation for the Poisson random measure.