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–Jacobi–Bellman type with the 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 ) ∈ Δ × 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.