Reflected backward stochastic differential equations under stopping with an arbitrary random time

Abstract

This paper addresses reflected backward stochastic differential equations (RBSDE hereafter) that take the form of   Here τ is an arbitrary random time that might not be a stopping time for the filtration F generated by the Brownian motion W . We consider the filtration G resulting from the progressive enlargement of F with τ where this becomes a stopping time, and study the RBSDE under G. Precisely, we focus on answering the following problems: a) What are the sufficient minimal conditions on the data (f, ξ, S, τ ) that guarantee the existence of the solution of the G-RBSDE in L p (p > 1)? b) How can we estimate the solution in norm using the triplet-data (f, ξ, S)? c) Is there an RBSDE under F that is intimately related to the current one and how their solutions are related to each other? This paper answers all these questions deeply and beyond. Importantly, we prove that for any random time, having a positive Azéma supermartingale, there exists a positive discount factor E -a positive and non-increasing F-adapted and RCLL process-that is vital in answering our questions without assuming any further assumption on τ , and determining the space for the triplet-data (f, ξ, S) and the space for the solution of the RBSDE as well. Furthermore, we found that the conditions for the G-RBSDE are weaker that the conditions for its F-RBSDE counterpart when the horizon is unbounded. Our approach sounds novel and very robust, as it relies on sharp martingale inequalities that hold no matter what is the filtration, and it treats both the linear and general case of RBSDEs for bounded and unbounded horizon. * Tahir Choulli is grateful to Polytechnique for the hospitality, where this work started in March 2018, and he is grateful to Nizar Touzi for introducing him to the RBSDEs and proposing him this project and its main ideas.

Optimal stopping problem under random horizon: Mathematical structures and linear RBSDEs

“…Thus, from this perspective and for the linear case, our RBSDE generalizes [39] to the cases where the horizon is infinite and having reflection barrier, while it generalizes [30] by letting S and K to be arbitrary RCLL. However, the nonlinear setting of [39] and [30], in which the driver f depends on Y and/or Z, can be found in [3,6].…”

“…Herein, we address (1.2) with τ being an arbitrary random time, while keeping the driver f independent of (Y, Z). For the general case of f , in this general setting of τ , we refer the reader to [3,6], and for the financial motivation of this sort of RBSDE we refer to [4,8,36,54] and the references therein to cite a few. In this general setting of τ and linear RBSDE, we treat three main problems and the challenges induced by them: a) What are the conditions (the weakest possible) on the data-triplet (f, S, ξ) that guarantee the existence and uniqueness of the solution to this RBSDE?…”

The Optimal Stopping Problem under a Random Horizon

“…The two processes Y − and S − are the left limits of Y and S, respectively, which are defined in Section 2.1 for the sake of a smooth presentation. For this direct application, we refer the reader to our earlier version of the complete work, which can be found in [7]. The relationship between the optimal stopping problem and RBSDEs is well understood nowadays, and we refer the reader to [8][9][10][11][12][13][14][15] and the references therein, to cite a few.…”