[Paper Review] Non linear optimal stopping problem and Reflected BSDEs in the predictable setting

Siham Bouhadou, Youssef Ouknine|arXiv (Cornell University)|Nov 2, 2018

Stochastic processes and financial applicationsEconomics, Econometrics and Finance19 references3 citations

TL;DR

This paper establishes the existence and uniqueness of reflected backward stochastic differential equations (RBSDEs) with predictable obstacles and non-quasi-left-continuous filtrations, using optimal stopping theory and Mertens' decomposition. It introduces a nonlinear optimal stopping problem under predictable $g$-expectations, proving the existence of an optimal predictable stopping time and characterizing the value function via the first component of the RBSDE.

ABSTRACT

In the first part of this paper, we study RBSDEs in the case where the filtration is not quasi-left continuous and the lower obstacle is given by a predictable process. We prove the existence and uniqueness by using some results of optimal stopping theory in the predictable setting, some tools from the general theory of processes as the Merten's decomposition of predictable strong supermartingale. In the second part we introduce an optimal stopping problem indexed by predictable stopping times with the non linear predictable $g$ expectation induced by an appropriate BSDE. We establish some useful properties of ${\cal{E}}^{p,g}$-supremartingales. Moreover, we show the existence of an optimal predictable stopping time, and we characterize the predictable value function in terms of the first component of RBSDEs studied in the first part.

Motivation & Objective

• To study reflected BSDEs when the filtration is not quasi-left-continuous and the obstacle is predictable.
• To establish existence and uniqueness of solutions using optimal stopping theory and general theory of processes.
• To introduce a nonlinear optimal stopping problem indexed by predictable stopping times under a $g$-expectation framework.
• To characterize the predictable value function using the first component of the RBSDE.
• To prove the existence of an optimal predictable stopping time in the context of $\mathcal{E}^{p,g}$-supremartingales.

Proposed method

• Utilizes Mertens' decomposition of predictable strong supermartingales to analyze the RBSDE structure.
• Applies tools from general theory of processes, including optional and predictable projections.
• Defines a predictable $g$-conditional expectation via a backward SDE with left-limits of martingales.
• Constructs a family of $\mathcal{E}^{p,g}$-supremartingales and derives their key properties.
• Uses the Snell envelope in the predictable setting to characterize the value function.
• Establishes the link between the optimal stopping problem and the solution of the RBSDE through the first component $Y$.

Experimental results

Research questions

• RQ1 How can RBSDEs be formulated and solved when the filtration is not quasi-left-continuous and the obstacle is predictable?
• RQ2 What are the necessary and sufficient conditions for the existence and uniqueness of solutions to such RBSDEs?
• RQ3 Can a nonlinear optimal stopping problem be defined using predictable $g$-expectations, and does it admit an optimal predictable stopping time?
• RQ4 How is the predictable value function related to the solution of the RBSDE?
• RQ5 What properties do $\mathcal{E}^{p,g}$-supremartingales satisfy in this framework?

Key findings

• The existence and uniqueness of solutions to the RBSDE with a predictable obstacle are established under a Lipschitz condition on the driver $g$.
• The predictable $g$-conditional expectation is well-defined and induces a family of $\mathcal{E}^{p,g}$-supremartingales.
• An optimal predictable stopping time exists for the nonlinear optimal stopping problem formulated with $\mathcal{E}^{p,g}$-supremartingales.
• The predictable value function is characterized as the first component $Y_0$ of the solution to the RBSDE.
• The solution $Y$ of the RBSDE is shown to be a predictable strong supermartingale, and its Mertens decomposition is used to prove uniqueness.
• The framework extends classical optimal stopping and BSDE theory to non-quasi-left-continuous settings, enabling modeling of predictable jumps in financial and stochastic control contexts.