Obliquely reflected backward stochastic differential equations

Abstract

In this paper, we study existence and uniqueness to multidimensional Reflected Backward Stochastic Differential Equations in an open convex domain, allowing for oblique directions of reflection. In a Markovian framework, combining a priori estimates for penalised equations and compactness arguments, we obtain existence results under quite weak assumptions on the driver of the BSDEs and the direction of reflection, which is allowed to depend on both Y and Z. In a non Markovian framework, we obtain existence and uniqueness result for direction of reflection depending on time and Y. We make use in this case of stability estimates that require some smoothness conditions on the domain and the direction of reflection.

Diagonally quadratic BSDE with oblique reflection and optimal switching

“…In consideration of linear reflections, Chassagneux et al [4] further obtained the existence and uniquess of the solution without the monotonicity condition. Chassagneux and Richou [5] futher studied existence and uniqueness of solutions for multidimensional RBSDEs in a non-empty open convex domain. More recently, Bénézet et al [1] considered a new class of obliquely reflected BSDEs which is related to • H p (R n×d ) the set of predictable R n×d -valued processes Z such that…”

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].…”

Systems of BSDES with oblique reflection and related optimal switching problems

“…In particular, only a Brownian filtration is considered. An interesting, different approach to optimal switching problem with constants switching costs and related reflected BSDEs is presented in the recent paper by Chassagneux and Richou [2]. As a matter of fact, in [2] much more general than in [8,9,10] equations with Brownian filtration and oblique reflection are considered.…”

“…An interesting, different approach to optimal switching problem with constants switching costs and related reflected BSDEs is presented in the recent paper by Chassagneux and Richou [2]. As a matter of fact, in [2] much more general than in [8,9,10] equations with Brownian filtration and oblique reflection are considered. Klimsiak [12] studied (1.1) without upper barrier, with general filtration F, L 1 -data and quasi-monotone generator f .…”