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October 2017 Reflected BSDEs when the obstacle is not right-continuous and optimal stopping
Miryana Grigorova, Peter Imkeller, Elias Offen, Youssef Ouknine, Marie-Claire Quenez
Ann. Appl. Probab. 27(5): 3153-3188 (October 2017). DOI: 10.1214/17-AAP1278


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.


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Miryana Grigorova. Peter Imkeller. Elias Offen. Youssef Ouknine. Marie-Claire Quenez. "Reflected BSDEs when the obstacle is not right-continuous and optimal stopping." Ann. Appl. Probab. 27 (5) 3153 - 3188, October 2017.


Received: 1 July 2015; Revised: 1 August 2016; Published: October 2017
First available in Project Euclid: 3 November 2017

zbMATH: 1379.60045
MathSciNet: MR3719955
Digital Object Identifier: 10.1214/17-AAP1278

Primary: 60G40 , 60H30 , 93E20
Secondary: 47N10‎ , 60G07

Keywords: $f$-expectation , backward stochastic differential equation , dynamic risk measure , Mertens’ decomposition , Optimal stopping , reflected backward stochastic differential equation , strong $\mathcal{E}^{f}$-supermartingale , strong optional supermartingale

Rights: Copyright © 2017 Institute of Mathematical Statistics


Vol.27 • No. 5 • October 2017
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