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2016 Generalized Dynkin games and doubly reflected BSDEs with jumps
Roxana Dumitrescu, Marie-Claire Quenez, Agnès Sulem
Electron. J. Probab. 21: 1-32 (2016). DOI: 10.1214/16-EJP4568

Abstract

We introduce a game problem which can be seen as a generalization of the classical Dynkin game problem to the case of a nonlinear expectation ${\cal E}^g$, induced by a Backward Stochastic Differential Equation (BSDE) with jumps with nonlinear driver $g$. Let $\xi , \zeta $ be two RCLL adapted processes with $\xi \leq \zeta $. The criterium is given by \[ {\cal J}_{\tau , \sigma }= {\cal E}^g_{0, \tau \wedge \sigma } \left (\xi _{\tau }\textbf{1} _{\{ \tau \leq \sigma \}}+\zeta _{\sigma }\textbf{1} _{\{\sigma <\tau \}}\right ), \] where $\tau $ and $ \sigma $ are stopping times valued in $[0,T]$. Under Mokobodzki’s condition, we establish the existence of a value function for this game, i.e. $\inf _{\sigma }\sup _{\tau } {\cal J}_{\tau , \sigma } = \sup _{\tau } \inf _{\sigma } {\cal J}_{\tau , \sigma }$. This value can be characterized via a doubly reflected BSDE. Using this characterization, we provide some new results on these equations, such as comparison theorems and a priori estimates. When $\xi $ and $\zeta $ are left upper semicontinuous along stopping times, we prove the existence of a saddle point. We also study a generalized mixed game problem when the players have two actions: continuous control and stopping. We then study the generalized Dynkin game in a Markovian framework and its links with parabolic partial integro-differential variational inequalities with two obstacles.

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Roxana Dumitrescu. Marie-Claire Quenez. Agnès Sulem. "Generalized Dynkin games and doubly reflected BSDEs with jumps." Electron. J. Probab. 21 1 - 32, 2016. https://doi.org/10.1214/16-EJP4568

Information

Received: 21 September 2015; Accepted: 5 October 2016; Published: 2016
First available in Project Euclid: 25 October 2016

zbMATH: 1351.93170
MathSciNet: MR3580030
Digital Object Identifier: 10.1214/16-EJP4568

Subjects:
Primary: 47N10‎, 60J60, 93E20

JOURNAL ARTICLE
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