Electronic Journal of Probability

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

Miryana Grigorova, Peter Imkeller, Youssef Ouknine, and Marie-Claire Quenez

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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 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.

Article information

Source
Electron. J. Probab., Volume 23 (2018), paper no. 122, 38 pp.

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

Permanent link to this document
https://projecteuclid.org/euclid.ejp/1545102140

Digital Object Identifier
doi:10.1214/18-EJP225

Mathematical Reviews number (MathSciNet)
MR3896859

Zentralblatt MATH identifier
07021678

Subjects
Primary: 60G40: Stopping times; optimal stopping problems; gambling theory [See also 62L15, 91A60] 93E20: Optimal stochastic control 60H30: Applications of stochastic analysis (to PDE, etc.)
Secondary: 60G07: General theory of processes 47N10: Applications in optimization, convex analysis, mathematical programming, economics

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

Rights
Creative Commons Attribution 4.0 International License.

Citation

Grigorova, Miryana; Imkeller, Peter; Ouknine, Youssef; Quenez, Marie-Claire. Doubly Reflected BSDEs and $\mathcal{E} ^{{f}}$-Dynkin games: beyond the right-continuous case. Electron. J. Probab. 23 (2018), paper no. 122, 38 pp. doi:10.1214/18-EJP225. https://projecteuclid.org/euclid.ejp/1545102140


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