# Dynamic programming for modulated jump-diffusions

by Nuno Azevedo (GFMUL)

We will discuss a class of stochastic optimal control problems with deterministic finite horizon and state variable dynamics determined by modulated jump diffusions. We will consider two alternative, yet related, classes of processes modulating the dynamics: a Markov process and a Semi-Markov process, both with a finite state space. We will focus mainly on the derivation of generalized dynamic programming principles, as well as on the corresponding Hamilton-Jacobi-Bellman equations which turn out as partial integro- differential equations. Time allowing, the existence of viscosity solutions for this last class of equations will be discussed.