Calculus of Variations and Geometric Measure Theory

C. Jimenez - M. Quincampoix

Hamilton Jacobi Isaacs equations for Differential Games with asymmetric information on probabilistic initial condition

created by jimenez on 08 Sep 2017
modified on 12 Dec 2019


Accepted Paper

Inserted: 8 sep 2017
Last Updated: 12 dec 2019

Journal: JMAA
Year: 2017

ERRATUM: the proof of the comparison principle is false.


We investigate Hamilton Jacobi Isaacs equations associated to a two-players zero-sum differential game with incomplete information. The first player has complete information on the initial state of the game while the second player has only information of a - possibly uncountable - probabilistic nature: he knows a probability measure on the initial state. Such differential games with finite type incomplete information can be viewed as a generalization of the famous Aumann-Maschler theory for repeated games. The main goal and novelty of the present work consists in obtaining and investigating a Hamilton Jacobi Isaacs Equation satisfied by the upper and the lower values of the game. Since we obtain a uniqueness result for such Hamilton Jacobi equation, as a byproduct, this gives an alternative proof of the existence of a value of the differential game (which has been already obtained in the literature by different technics). Since the Hamilton Jacobi equation is naturally stated in the space of probability measures, we use the Wasserstein distance and some tools of optimal transport theory.