## The Annals of Probability

- Ann. Probab.
- Volume 42, Number 4 (2014), 1724-1768.

### Value in mixed strategies for zero-sum stochastic differential games without Isaacs condition

Rainer Buckdahn, Juan Li, and Marc Quincampoix

#### 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.

#### Article information

**Source**

Ann. Probab., Volume 42, Number 4 (2014), 1724-1768.

**Dates**

First available in Project Euclid: 3 July 2014

**Permanent link to this document**

https://projecteuclid.org/euclid.aop/1404394077

**Digital Object Identifier**

doi:10.1214/13-AOP849

**Mathematical Reviews number (MathSciNet)**

MR3111671

**Zentralblatt MATH identifier**

1296.49034

**Subjects**

Primary: 49N70: Differential games 49L25: Viscosity solutions

Secondary: 91A23: Differential games [See also 49N70] 60H10: Stochastic ordinary differential equations [See also 34F05]

**Keywords**

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

#### Citation

Buckdahn, Rainer; Li, Juan; Quincampoix, Marc. Value in mixed strategies for zero-sum stochastic differential games without Isaacs condition. Ann. Probab. 42 (2014), no. 4, 1724--1768. doi:10.1214/13-AOP849. https://projecteuclid.org/euclid.aop/1404394077