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

Miryana Grigorova (1) , Peter Imkeller (2) , Youssef Ouknine (3) , Marie-Claire Quenez (4)

1 KVW - Centre for Risk and Insurance, Hanover
2 Institut für Mathematik [Humboldt]
3 [MRC-CAD] - Department of Mathematics [Marrakech]
4 LPMA - Laboratoire de Probabilités et Modèles Aléatoires

We formulate a notion of doubly reflected BSDE in the case where the barriers $\xi$ and $\zeta$ do not satisfy any regularity assumption. Under a technical assumption (a Mokobodzki-type condition), we show existence and uniqueness of the solution. In the case where $\xi$ and $-\zeta$ are assumed to be right-uppersemicontinuous, 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.

Mots clés: Dynkin game; Doubly reflected BSDEs; backward stochastic differential equations; saddle points; $f$-expectation; nonlinear expectation; game option

Domaines: Probabilités [math.PR]; Optimisation et contrôle [math.OC]; Finance quantitative [q-fin.CP]

Miryana Grigorova, Peter Imkeller, Youssef Ouknine, Marie-Claire Quenez. Doubly Reflected BSDEs and ${\cal E}^{f}$-Dynkin games: beyond the right-continuous case. 2017. ⟨hal-01497914v1⟩