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

Abstract

We introduce a new formulation of reflected BSDEs and doubly reflected BSDEs associated with irregular obstacles. In the first part of the paper, we consider an extension of the classical optimal stopping problem over a larger set of stopping systems than the set of stopping times (namely, the set of split stopping times), where the payoff process ξ is irregular and in the case of a general filtration. We show that the value family can be aggregated by an optional process v, which is characterized as the Snell envelope of the reward process ξ over split stopping times. Using this, we prove the existence and uniqueness of a solution Y to irregular reflected BSDEs. In the second part of the paper, motivated by the classical Dynkin game with completely irregular rewards considered by Grigorova et al. (2018), we generalize the previous equations to the case of two reflecting barrier processes. Under a general type of Mokobodzki's condition, we show the existence of the solution through a Picard iteration method and a Banach fixed point theorem.