Please use this identifier to cite or link to this item: http://hdl.handle.net/10397/5879
Title: Regularity properties for general HJB equations : a backward stochastic differential equation method
Authors: Buckdahn, R
Huang, J 
Li, J
Keywords: Backward stochastic differential equation
HJB equation
Lipschitz continuity
Reflected backward stochastic differential equations
Semiconcavity
Value function
Issue Date: 2012
Publisher: Society for Industrial and Applied Mathematics
Source: SIAM journal on control and optimization, 2012, v. 50, no. 3, p. 1466–1501 How to cite?
Journal: SIAM Journal on control and optimization 
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.
URI: http://hdl.handle.net/10397/5879
ISSN: 0363-0129
EISSN: 1095-7138
DOI: 10.1137/110828629
Rights: © 2012 Society for Industrial and Applied Mathematics
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