Reflected BSDEs when the obstacle is not right-continuous and optimal stopping

Miryana Grigorova, Peter Imkeller, Elias Offen, Youssef Ouknine and Marie-Claire Quenez
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Miryana Grigorova: Institut für Mathematik [Humboldt] - HU Berlin - Humboldt-Universität zu Berlin = Humboldt University of Berlin = Université Humboldt de Berlin
Peter Imkeller: Institut für Mathematik [Humboldt] - HU Berlin - Humboldt-Universität zu Berlin = Humboldt University of Berlin = Université Humboldt de Berlin
Elias Offen: UB - University of Botswana
Youssef Ouknine: UCA - Université Cadi Ayyad [Marrakech]
Marie-Claire Quenez: LPMA - Laboratoire de Probabilités et Modèles Aléatoires - UPMC - Université Pierre et Marie Curie - Paris 6 - UPD7 - Université Paris Diderot - Paris 7 - CNRS - Centre National de la Recherche Scientifique

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Abstract: 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 tools from the general theory of processes such as Mertens decomposition of optional strong (but not necessarily right-continuous) supermartingales, some tools from optimal stopping theory, 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 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.

Keywords: Mertens decomposition; reflected backward stochastic differential equation; optimal stopping; dynamic risk measure; f -expectation; strong optional supermartingale; backward stochastic differential equation (search for similar items in EconPapers)
Date: 2017
Note: View the original document on HAL open archive server: https://hal.science/hal-01141801v2
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Citations: View citations in EconPapers (16)

Published in The Annals of Applied Probability, 2017

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Persistent link: https://EconPapers.repec.org/RePEc:hal:journl:hal-01141801

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