Zero-sum path-dependent stochastic differential games in weak formulation

We consider zero-sum stochastic differential games with possibly path-dependent volatility controls. Unlike the previous literature, we allow for weak solutions of the state equation so that the players’ controls are automatically of feedback type. In particular, we do not require the controls to be “simple,” which has fundamental importance for the possible existence of saddle-points. Under some restrictions, needed for the a priori regularity of the upper and lower value functions of the game, we show that the game value exists when both the appropriate path-dependent Isaacs condition, and the uniqueness of viscosity solutions of the corresponding path-dependent Isaacs-HJB equation hold. We also provide a general verification argument and a characterisation of saddle-points by means of an appropriate notion of second-order backward SDE.