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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 |
Issue Date: | 2012 | Source: | SIAM journal on control and optimization, 2012, v. 50, no. 3, p. 1466–1501 | 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. | Keywords: | Backward stochastic differential equation HJB equation Lipschitz continuity Reflected backward stochastic differential equations Semiconcavity Value function |
Publisher: | Society for Industrial and Applied Mathematics | Journal: | SIAM Journal on control and optimization | ISSN: | 0363-0129 | EISSN: | 1095-7138 | DOI: | 10.1137/110828629 | Rights: | © 2012 Society for Industrial and Applied Mathematics |
Appears in Collections: | Journal/Magazine Article |
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