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

Abstract

We consider an optimal control problem for piecewise deterministic Markov processes (PDMPs) on a bounded state space. The control problem under study is very general: a pair of controls acts continuously on the deterministic flow and on the two transition measures (in the interior and from the boundary of the domain) 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 fully-nonlinear Hamilton-Jacobi-Bellman equation with a non-local type boundary condition.By means of the recent control randomization method, we are able to provide a probabilistic representation for the value function in terms of a constrained backward stochastic differential equation (BSDE), known as nonlinear Feynman-Kac formula. This result considerably extends the existing literature, where only the case with no jumps from the boundary is considered. The additional boundary jump mechanism is described in terms of a non quasi-left-continuous random measure and induces predictable jumps in the PDMP's dynamics. The existence and uniqueness results for BSDEs driven by such a random measure are non trivial, even in the unconstrained case, as emphasized in the recent work [2].Keywords: Backward stochastic differential equations, optimal control problems, piecewise deterministic Markov processes, non quasi-left-continuous random measure, randomization of controls.MSC 2010: 93E20, 60H10, 60J25.

Stochastic filtering and optimal control of pure jump Markov processes with noise-free partial observation

“…We also note an important difference between the approach to PDP optimal control problems presented in [32] (and employed in other works as [1,5,6,26,30,31,33,46]) and ours. In the book by Davis the class of control processes is represented by piecewise open-loop controls, introduced by D. Vermes in [49].…”

Optimal Control of Infinite-Dimensional Piecewise Deterministic Markov Processes: A BSDE Approach. Application to the Control of an Excitable Cell Membrane

“…Many generalizations of the present work may be possible. For instance, it would be interesting to treat the general case with infinite-dimensional PDMPs on a state space with boundary, from which additional instantaneous jumps into the interior of the domain may occur (in the finite-dimensional case, this feature has been recently considered in [3]). Moreover, in our application section we have considered the classical case of a Laplacian operator, but other operators could be addressed as well.…”

The identification problem for BSDEs driven by possibly non quasi-left-continuous random measures

“…Existence and uniqueness results for solutions (Y, U ) ∈ L 2 × G 2 (µ) to BSDEs driven by purely discontinuous martingales (that include (5.25) as a special case) were established under suitable assumptions in [1] and in the recent work [3].…”