Regularity properties for general HJB equations: A backward stochastic differential equation method

Rainer Buckdahn, Jianhui Huang, Juan Li

Research output: Journal article publicationJournal articleAcademic researchpeer-review

8 Citations (Scopus)

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 the form of a stochastic control system in which nonlinear cost functional is defined with the help of a backward stochastic differential equation (BSDE) or a reflected BSDE. More precisely, we prove that, first, the unique viscosity solution V (t, x) of an 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). Second, for the case that V solves an 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 [Nonlinear Differential Equations Appl., 17 (2010), pp. 715-728]. Our approach embeds their idea of time change into a BSDE analysis. We also provide an elementary counterexample which shows that, in general, for the case that V solves an HJB equation with a lower obstacle the semiconcavity doesn't hold true.
Original languageEnglish
Pages (from-to)1466-1501
Number of pages36
JournalSIAM Journal on Control and Optimization
Volume50
Issue number3
DOIs
Publication statusPublished - 3 Sep 2012

Keywords

  • Backward stochastic differential equation
  • HJB equation
  • Lipschitz continuity
  • Reflected backward stochastic differential equations
  • Semiconcavity
  • Value function

ASJC Scopus subject areas

  • Control and Optimization
  • Applied Mathematics

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