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

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.