Constrained BSDEs Driven by a Non-Quasi-Left-Continuous Random Measure and Optimal Control of PDMPs on Bounded Domains

摘要

We consider an optimal control problem for piecewise deterministic Markovprocesses (PDMPs) on a bounded state space. The control problem under study isvery general: a pair of controls acts continuously on the deterministic flowand on the two transition measures (in the interior and from the boundary ofthe domain) describing the jump dynamics of the process. For this class ofcontrol problems, the value function can be characterized as the uniqueviscosity solution to the corresponding fully-nonlinear Hamilton-Jacobi-Bellmanequation with a non-local type boundary condition. By means of the recentcontrol randomization method, we are able to provide a probabilisticrepresentation for the value function in terms of a constrained backwardstochastic differential equation (BSDE), known as nonlinear Feynman-Kacformula. This result considerably extends the existing literature, where onlythe case with no jumps from the boundary is considered. The additional boundaryjump mechanism is described in terms of a non quasi-left-continuous randommeasure and induces predictable jumps in the PDMP's dynamics. The existence anduniqueness results for BSDEs driven by such a random measure are non trivial,even in the unconstrained case, as emphasized in the recent work [2].