# Continuous-Time Dynkin Games with Mixed Strategies

Abstract : Let (X,Y,Z) be a triple of payoff processes defining a Dynkin game \tilde R(\sigma,\tau) &=& E\left[ X_\sigma\1_{\{\tau > \sigma\}} +Y_\tau \1_{\{\tau < \sigma\}} +Z_\tau \1_{\{\tau=\sigma\}}\right] , where $\sigma$ and $\tau$ are stopping times valued in [0,T]. In the case Z=Y, it is well known that the condition X $\leq$ Y is needed in order to establish the existence of value for the game, i.e., $\inf_{\tau}\sup_{\sigma}\tilde R(\sigma,\tau)$ $=$ $\sup_{\sigma}\inf_{\tau}\tilde R(\sigma,\tau)$. In order to remove the condition $X$ $\leq$ $Y$, we introduce an extension of the Dynkin game by allowing for an extended set of strategies, namely, the set of mixed strategies. The main result of the paper is that the extended Dynkin game has a value when $Z\leq Y$, and the processes X and Y are restricted to be semimartingales continuous at the terminal time T.
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https://hal-hec.archives-ouvertes.fr/hal-00465013
Contributor : Antoine Haldemann <>
Submitted on : Thursday, March 18, 2010 - 4:08:56 PM
Last modification on : Thursday, January 11, 2018 - 6:19:31 AM

### Citation

Nicolas Vieille, Nizar Touzi. Continuous-Time Dynkin Games with Mixed Strategies. SIAM Journal on Control and Optimization, Society for Industrial and Applied Mathematics, 2002, Vol.41,n°4, pp.1073-1088. ⟨10.1137/S0363012900369812⟩. ⟨hal-00465013⟩

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