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2018 Doubly Reflected BSDEs and $\mathcal{E} ^{{f}}$-Dynkin games: beyond the right-continuous case
Miryana Grigorova, Peter Imkeller, Youssef Ouknine, Marie-Claire Quenez
Electron. J. Probab. 23: 1-38 (2018). DOI: 10.1214/18-EJP225


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 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 $\boldsymbol{\mathcal {E}} ^f$-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 a priori estimates with universal constants.


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Miryana Grigorova. Peter Imkeller. Youssef Ouknine. Marie-Claire Quenez. "Doubly Reflected BSDEs and $\mathcal{E} ^{{f}}$-Dynkin games: beyond the right-continuous case." Electron. J. Probab. 23 1 - 38, 2018.


Received: 17 May 2017; Accepted: 14 September 2018; Published: 2018
First available in Project Euclid: 18 December 2018

zbMATH: 07021678
MathSciNet: MR3896859
Digital Object Identifier: 10.1214/18-EJP225

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

Keywords: $f$-expectation , Backward stochastic differential equations , cancellable American option , doubly reflected BSDEs , Dynkin game , Game option , general filtration , nonlinear expectation , saddle points , stopping system , stopping time

Vol.23 • 2018
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