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July 2014 Value in mixed strategies for zero-sum stochastic differential games without Isaacs condition
Rainer Buckdahn, Juan Li, Marc Quincampoix
Ann. Probab. 42(4): 1724-1768 (July 2014). DOI: 10.1214/13-AOP849

Abstract

In the present work, we consider 2-person zero-sum stochastic differential games with a nonlinear pay-off functional which is defined through a backward stochastic differential equation. Our main objective is to study for such a game the problem of the existence of a value without Isaacs condition. Not surprising, this requires a suitable concept of mixed strategies which, to the authors’ best knowledge, was not known in the context of stochastic differential games. For this, we consider nonanticipative strategies with a delay defined through a partition $\pi$ of the time interval $[0,T]$. The underlying stochastic controls for the both players are randomized along $\pi$ by a hazard which is independent of the governing Brownian motion, and knowing the information available at the left time point $t_{j-1}$ of the subintervals generated by $\pi$, the controls of Players 1 and 2 are conditionally independent over $[t_{j-1},t_{j})$. It is shown that the associated lower and upper value functions $W^{\pi}$ and $U^{\pi}$ converge uniformly on compacts to a function $V$, the so-called value in mixed strategies, as the mesh of $\pi$ tends to zero. This function $V$ is characterized as the unique viscosity solution of the associated Hamilton–Jacobi–Bellman–Isaacs equation.

Citation

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Rainer Buckdahn. Juan Li. Marc Quincampoix. "Value in mixed strategies for zero-sum stochastic differential games without Isaacs condition." Ann. Probab. 42 (4) 1724 - 1768, July 2014. https://doi.org/10.1214/13-AOP849

Information

Published: July 2014
First available in Project Euclid: 3 July 2014

zbMATH: 1296.49034
MathSciNet: MR3111671
Digital Object Identifier: 10.1214/13-AOP849

Subjects:
Primary: 49L25 , 49N70
Secondary: 60H10 , 91A23

Keywords: 2-person zero-sum stochastic differential game , Backward stochastic differential equations , dynamic programming principle , Isaacs condition , randomized controls , value function , viscosity solution

Rights: Copyright © 2014 Institute of Mathematical Statistics

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Vol.42 • No. 4 • July 2014
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