Reflected backward stochastic differential equations with jumps in time-dependent random convex domains

Abstract

In this paper, we study a class of multi-dimensional reflected backward stochastic differential equations when the noise is driven by a Brownian motion and an independent Poisson point process, and when the solution is forced to stay in a time-dependent adapted and continuous convex domainWe prove the existence an uniqueness of the solution, and we also show that the solution of such equations may be approximated by backward stochastic differential equations with jumps reflected in appropriately defined discretizations of D, via a penalization method. 1 case of fixed convex domains were for the first time studied in [3]. Actually, the authors study multidimensional RBSDEs in the case of fixed convex domain C = {C t , t ∈ [0, T ]}, of the form:where K t is continuous, increasing and of bounded total variation |K| satisfying K 0 = 0. The last condition insures that K is minimal in the sense that it increases only when Y is at the boundary of C. In fact, the process K is inward normal to C at Y , precisely K t = t 0 η s d|K| s such that η s ∈ N (Y s ) and where N (Y s ) is the inward normal unit vector to C at Y s . Actually, when Y is at the boundary it is pushed into the domain along η ∈ N (Y ). The authors provide existence and uniqueness for such RBSDEs via a penalization method. Later, [12] extended the result of [3] to the case of jumps (i.e. whose noise includes a Poisson random measure part). The author studied RBSDE of Wiener-Poisson type in fixed convex domain C, for which he established existence and uniqueness using a penalization method. They considered RBSDEs of the following form:• M: the space of IR m×d -valued, F-progressively measurable processes (Z t ) 0≤t≤T such that T 0 Z s 2 ds < ∞ P-a.s, and equipped with the metric θ(Z, Z ′ ) = E T 0 Z s − Z ′ s 2 ds ∧ 1 . • L: the set of mappings V : Ω × [0, T ] × U → IR m which are P ⊗ U -measurable, such that T 0 U |V s (e)| 2 λ(de)ds < ∞ P-a.s, and equipped with the metric ̟(V, V ′ ) = E T 0 U |V s (e) − V ′ s (e)| 2 λ(de)ds ∧ 1 .

Reflected BSDEs in non-convex domains

“…[13,9,12,21,22]. However, the multidimensional case presents significant additional challenges (e.g., due to the lack of the comparison principle), and, to date, the well-posedness of multidimensional reflected BSDEs (or, systems of reflected BSDEs) has only been established in the case of convex reflection domains: see, e.g., [20,32,8,17]. The systems of reflected BSDEs in convex domains appear in certain types of stochastic control problems, such as the switching problems: see, among others, [24,27,7,6,35,1].…”