BSDEs driven by a general random measure and optimal control for piecewise deterministic Markov processes”

Abstract 

We consider an optimal control problem for piecewise deterministic Markov processes (PDMP) on
a bounded state space. Here a pair of controls acts continuously on the deterministic flow and on
the transition measure describing the jump dynamics of the process. For this class of control
problems, the value function can be characterized as the unique viscosity solution to the
corresponding integro-differential Hamilton-Jacobi-Bellman equation with a non-loca type
boundary condition. We are able to provide a probabilistic representation for the value function in
terms of a suitable backward stochastic differential equation, known as nonlinear Feynman-Kac
formula. The jump mechanism from the boundary entails the presence of predictable jumps in the
PDMP dynamics, so that the associated BSDE turns out to be driven by a random measure with
predictable jumps. Existence and uniqueness results for such a class of equations are non-trivial
and are related to recent works on well-posedness for BSDEs driven by non quasi-left-continuous
random measures.

L’ Organizzatore                                      Il Direttore
Dott. Alberto Lanconelli                            Prof. Angela Montanari