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Regularity Properties of Viscosity Solutions of Integro-Partial Differential Equations of Hamilton-Jacobi-Bellman Type

We study the regularity properties of integro-partial differential equations of Hamilton-Jocobi-Bellman type with terminal condition, which can be interpreted through a stochastic control system, composed of a forward and a backward stochastic differential equation, both driven by a Brownian motion and a compensated Poisson random measure. More precisely, we prove that, under appropriate assumptions, the viscosity solution of such equations is jointly Lipschitz and jointly semiconcave in $(t,x)\inΔ\times\R^d$, for all compact time intervals $Δ$ excluding the terminal time. Our approach is based on the time change for the Brownian motion and on Kulik's transformation for the Poisson random measure.