A probabilistic representation for the value of zero-sum differential games with incomplete information on both sides

TL;DR

This paper establishes a probabilistic representation for the value of zero-sum differential games with incomplete information on both sides, linking it to stochastic differential games with complete information and continuous martingale controls.

Contribution

It introduces a novel probabilistic representation for these complex games, extending previous models to continuous-time and unbounded control spaces, and connects to Hamilton-Jacobi equations with convexity constraints.

Findings

Value of the game is represented as a stochastic differential game with complete information.

Provides a new probabilistic interpretation of Hamilton-Jacobi equations with convexity constraints.

Extends the splitting-game concept to continuous-time models with unbounded controls.

Abstract

We prove that for a class of zero-sum differential games with incomplete information on both sides, the value admits a probabilistic representation as the value of a zero-sum stochastic differential game with complete information, where both players control a continuous martingale. A similar representation as a control problem over discontinuous martingales was known for games with incomplete information on one side (see Cardaliaguet-Rainer [8]), and our result is a continuous-time analog of the so called splitting-game introduced in Laraki [20] and Sorin [27] in order to analyze discrete-time models. It was proved by Cardaliaguet [4, 5] that the value of the games we consider is the unique solution of some Hamilton-Jacobi equation with convexity constraints. Our result provides therefore a new probabilistic representation for solutions of Hamilton-Jacobi equations with convexity constraints as values of stochastic differential games with unbounded control spaces and unbounded volatility.