Electronic Journal of Probability

Generalized Dynkin games and doubly reflected BSDEs with jumps

Roxana Dumitrescu, Marie-Claire Quenez, and Agnès Sulem

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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.

Article information

Electron. J. Probab., Volume 21 (2016), paper no. 64, 32 pp.

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

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Primary: 93E20: Optimal stochastic control 60J60: Diffusion processes [See also 58J65] 47N10: Applications in optimization, convex analysis, mathematical programming, economics

Dynkin game mixed game problem nonlinear expectation $g$-evaluation doubly reflected BSDEs partial integro-differential variational inequalities game option

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Dumitrescu, Roxana; Quenez, Marie-Claire; Sulem, Agnès. Generalized Dynkin games and doubly reflected BSDEs with jumps. Electron. J. Probab. 21 (2016), paper no. 64, 32 pp. doi:10.1214/16-EJP4568. https://projecteuclid.org/euclid.ejp/1477395747

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