This is a joint work with Alessandro Bondi (Luiss). We prove a Dynamic Programming Principle (DPP) in a strong formulation for a stochastic control problem involving controlled SDEs with jumps driven by a Brownian motion and a stationary Poisson point process with values in R^d. We consider arbitrary predictable controls a with values in a closed convex set C subset R^l. The coefficients of the SDE satisfy linear growth and Lipschitz-type conditions in the x-variable, and are continuous in the control variable. Moreover, we deal with the value function v(s,x)= sup_{a} E}{\int_{s}^{T} h (r,X_r^{s,x,a}, a_r )dr + j(X_T^{s,x,a}) }, assuming that h and j are bounded and continuous; here X_r^{s,x,a} is the solution to the controlled SDE. To prove the DPP we show the existence of a regular stochastic flow for the SDEs when the coefficients are independent of the control a. Notably, this regularity result is new for jump diffusions even when there is no large-jumps component (cf. Kunita's recent monograph on stochastic flows and jump diffusions). The proof of the DPP is completed by introducing an approach that relies on a suitable subclass of finitely generated step controls. These controls allow us to apply a basic measurable selection theorem by L. D. Brown and R. Purves; we believe that this novel method is of independent interest even in the Brownian case without jumps.

TBA