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

Fabien Gensbittel (1) , Catherine Rainer (2)

1 GREMAQ - Groupe de recherche en économie mathématique et quantitative
2 LM - Laboratoire de mathématiques de Brest

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.

Mots clés: stochastic differential game; Hamilton-Jacobi equations; zerosum games; incomplete information

Domaines: Optimisation et contrôle [math.OC]

Fabien Gensbittel, Catherine Rainer. A probabilistic representation for the value of zero-sum differential games with incomplete information on both sides. SIAM Journal on Control and Optimization, 2017, 55 (2), pp.693-723. ⟨hal-01275809v2⟩