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

TL;DR

This paper studies zero-sum stochastic differential games with path-dependent states using weak solutions, establishing conditions for the existence of game value and characterizing saddle-points via second-order backward SDEs.

Contribution

It introduces a framework allowing weak solutions in path-dependent stochastic games, ensuring value existence under Isaacs condition and viscosity solution uniqueness.

Findings

Game value exists under certain regularity and Isaacs condition.

Characterization of saddle-points via second-order backward SDEs.

Provides a verification method for saddle-points in path-dependent games.

Abstract

We consider zero-sum stochastic differential games with possibly path-dependent controlled state. Unlike the previous literature, we allow for weak solutions of the state equation so that the players' controls are automatically of feedback type. 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 SDEs.