Reflected BSDEs when the obstacle is not right-continuous and optimal stopping. (English) Zbl 1379.60045

Grigorova, Miryana; Imkeller, Peter; Offen, Elias; Ouknine, Youssef; Quenez, Marie-Claire

Ann. Appl. Probab. 27, No. 5, 3153-3188 (2017).

Summary: In the first part of the paper, we study reflected backward stochastic differential equations (RBSDEs) with lower obstacle which is assumed to be right upper-semicontinuous but not necessarily right-continuous. We prove existence and uniqueness of the solutions to such RBSDEs in appropriate Banach spaces. The result is established by using some results from optimal stopping theory, some tools from the general theory of processes such as Mertens’ decomposition of optional strong supermartingales, as well as an appropriate generalization of Itô’s formula due to Gal’chouk and Lenglart. In the second part of the paper, we provide some links between the RBSDE studied in the first part and an optimal stopping problem in which the risk of a financial position \(\xi\) is assessed by an \(f\)-conditional expectation \(\mathcal{E}^{f}(\cdot)\) (where \(f\) is a Lipschitz driver). We characterize the “value function” of the problem in terms of the solution to our RBSDE. Under an additional assumption of left upper-semicontinuity along stopping times on \(\xi\), we show the existence of an optimal stopping time. We also provide a generalization of Mertens’ decomposition to the case of strong \(\mathcal{E}^{f}\)-supermartingales.

MSC:

60G40	Stopping times; optimal stopping problems; gambling theory
60H30	Applications of stochastic analysis (to PDEs, etc.)
60G07	General theory of stochastic processes
47N10	Applications of operator theory in optimization, convex analysis, mathematical programming, economics
60G48	Generalizations of martingales

Keywords:

backward stochastic differential equation; reflected backward stochastic differential equation; optimal stopping; \(f\)-expectation; strong optional supermartingale; Mertens decomposition; dynamic risk measure; strong \(\mathcal{E}^{f}\)-supermartingale