Regularity properties for general HJB equations. A BSDE method

TL;DR

This paper establishes regularity properties, including Lipschitz continuity and semiconcavity, of solutions to a broad class of HJB equations using a BSDE approach, extending previous results and providing counterexamples.

Contribution

It introduces a BSDE-based method to prove regularity of HJB solutions, extending prior work and addressing cases with obstacles.

Findings

Viscosity solutions are jointly Lipschitz in (t,x).

Solutions without obstacles are jointly semiconcave in (t,x).

Counterexample shows semi-concavity fails with lower obstacles.

Abstract

In this work we investigate regularity properties of a large class of Hamilton-Jacobi-Bellman (HJB) equations with or without obstacles, which can be stochastically interpreted in form of a stochastic control system which nonlinear cost functional is defined with the help of a backward stochastic differential equation (BSDE) or a reflected BSDE (RBSDE). More precisely, we prove that, firstly, the unique viscosity solution $V(t,x)$ of such a HJB equation over the time interval $[0,T],$ with or without an obstacle, and with terminal condition at time $T$, is jointly Lipschitz in $(t,x)$, for $t$ running any compact subinterval of $[0,T)$. Secondly, for the case that $V$ solves a HJB equation without an obstacle or with an upper obstacle it is shown under appropriate assumptions that $V(t,x)$ is jointly semiconcave in $(t,x)$. These results extend earlier ones by Buckdahn, Cannarsa and Quincampoix [1]. Our approach embeds their idea of time change into a BSDE analysis. We also provide an elementary counter-example which shows that, in general, for the case that $V$ solves a HJB equation with a lower obstacle the semi-concavity doesn't hold true.