# Doubly Reflected BSDEs and ${\cal E}^{f}$-Dynkin games: beyond the right-continuous case

Miryana Grigorova (), Peter Imkeller, Youssef Ouknine and Marie-Claire Quenez ()
Miryana Grigorova: Bielefeld University
Peter Imkeller: Institut für Mathematik [Berlin] - TUB - Technische Universität Berlin
Youssef Ouknine: Faculté des Sciences Semlalia Marrakech
Marie-Claire Quenez: LPSM UMR 8001 - Laboratoire de Probabilités, Statistique et Modélisation - UPMC - Université Pierre et Marie Curie - Paris 6 - UPD7 - Université Paris Diderot - Paris 7 - CNRS - Centre National de la Recherche Scientifique

Abstract: We formulate a notion of doubly reflected BSDE in the case where the barriers $\xi$ and $\zeta$ do not satisfy any regularity assumption and with a general filtration. Under a technical assumption (a Mokobodzki-type condition), we show existence and uniqueness of the solution. In the case where $\xi$ is right upper-semicontinuous and $\zeta$ is right lower-semicontinuous, the solution is characterized in terms of the value of a corresponding $\mathcal{E}^f$-Dynkin game, i.e. a game problem over stopping times with (non-linear) $f$-expectation, where $f$ is the driver of the doubly reflected BSDE. In the general case where the barriers do not satisfy any regularity assumptions, the solution of the doubly reflected BSDE is related to the value of "an extension" of the previous non-linear game problem over a larger set of "stopping strategies" than the set of stopping times. This characterization is then used to establish a comparison result and \textit{a priori} estimates with universal constants.

Keywords: stopping time; stopping system; general filtration; cancellable Aperican option; game option; nonlinear expectation; f -expectation; saddle points; Doubly reflected BSDEs; backward stochastic differential equations; Dynkin game (search for similar items in EconPapers)
New Economics Papers: this item is included in nep-gth
Date: 2018-07-14
Note: View the original document on HAL open archive server: https://hal.archives-ouvertes.fr/hal-01497914v3
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