Doubly Reflected Backward Stochastic Differential Equations in the Predictable Setting

Reflected and Doubly RBSDEs with Irregular Obstacles and a Large Set of Stopping Strategies

“…Remark 4.2 In Definition 4.1, by considering δ = (∅, τ ), where τ runs through the set of stopping times T 0 , we recover the usual formulation of the doubly reflected BSDEs with optional obstacles given in [19]. Moreover, if we associate T with the split terminal time ρ T = (Ω, T ) and we consider δ = (Ω, τ ), where τ runs through the set of predictable stopping times T p 0 , our definition coincides with the definition of doubly reflected BSDEs for predictable obstacles given in [11]. where the process J g and Jg satisfy the following coupled system of reflected BSDEs: From Remark 4.5, J g and Jg are two nonnegative optional strong supermartingales in S 2 , hence of class (D) (i.e.…”

“…They show the existence of a unique solution if and only if the so-called Mokobodzki's condition holds (there exist two strong supermartingales such that their difference is between ξ and ζ). Recently, Arharas et al (2021) [11] generalized the work [33], using tools from the general theory of stochastic processes, for instance, Gal'chouk and Lenglart formula for strong predictable semimartingales and the predictable Mertens decomposition. They formulate a notion of DRBSDEs in the predictable setting, where the obstacles are predictable processes and the filtration is non-quasi-left continuous.…”