Two Person Zero-Sum Game in Weak Formulation and Path Dependent Bellman-Isaacs Equation

In this paper we study a two person zero sum stochastic differential game in weak formulation. Unlike the standard literature, which uses strategy type controls, the weak formulation allows us to consider the game with control against control. We shall prove the existence of game value under natural conditions. Another main feature of the paper is that we allow for non-Markovian structure, and thus the game value is a random process. We characterize the value process as the unique viscosity solution of the corresponding path dependent Bellman–Isaacs equation, a notion recently introduced by Ekren et al. [Ann. Probab., 42 (2014), pp. 204–236] and Ekren, Touzi, and Zhang [Stochastic Process., to appear; preprint, arXiv:1210.0006v2; preprint, arXiv:1210.0007v2].