Existence and uniqueness for backward stochastic differential equations driven by a random measure, possibly non quasi-left continuous

Abstract

We study the following backward stochastic differential equation on finite time horizon driven by an integer-valued random measure µ on R + ×E, where E is a Lusin space, with compensator ν(dt, dx) = dA t φ t (dx):The generator f satisfies, as usual, a uniform Lipschitz condition with respect to its last two arguments. In the literature, the existence and uniqueness for the above equation in the present general setting has only been established when A is continuous or deterministic. The general case, i.e. A is a right-continuous nondecreasing predictable process, is addressed in this paper. These results are relevant, for example, in the study of control problems related to Piecewise Deterministic Markov Processes (PDMPs). Indeed, when µ is the jump measure of a PDMP, then A is predictable (but not deterministic) and discontinuous, with jumps of size equal to 1.

Lp solutions of backward stochastic differential equations with jumps

“…As to BSDEs driven by other discontinuous random sources, Xia [72] and Bandini [6] studied BSDEs driven by a random measure; Confortola et al [25,26] considered BSDEs driven by a marked point process; [61,5,66,36] analyzed BSDEs driven by Lévy processes; [2,68,46] discussed BSDEs driven by a process with a finite number of marked jumps.…”

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

“…For such an equation, the proof of existence and uniqueness is a difficult task, and counterexamples can be obtained even in simple cases, see [14]. Only recently, some results in the unconstrained case have been obtained in this context, see [13], [12], [2]. In order to have an existence and uniqueness result for our BSDE, we have to impose the following additional assumption on p * .…”

“…In particular, the compensatorp has predictable jumpsp({t} × dy db dc) = 1 X t− ∈∂E . Equation (4.2)-(4.3)-(4.4) is thus driven by a non quasileft-continuous random measure; the associated well-posedness results are obtained by means of a penalization approach, by suitably extending the recent existence and uniqueness theorem obtained in [2] for unconstrained BSDEs. Once we achieve the existence and uniqueness of a maximal solution to (4.2)-(4.3)-(4.4), we prove that its component Y x,a 0 ,a Γ at the initial time represents the randomized value function, i.e.…”

“…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].…”

“…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.…”