Continuous-Time Markov Games with Asymmetric Information

Abstract

We study a two-player zero-sum stochastic differential game with asymmetric information where the payoff depends on a controlled continuous-time Markov chain X with finite state space which is only observed by player 1. This model was already studied in Cardaliaguet et al (Math Oper Res 41(1):49–71, 2016) through an approximating sequence of discrete-time games. Our first contribution is the proof of the existence of the value in the continuous-time model based on duality techniques. This value is shown to be the unique solution of the same Hamilton–Jacobi equation with convexity constraints which characterized the limit value obtained in Cardaliaguet et al. (2016). Our second main contribution is to provide a simpler equivalent formulation for this Hamilton–Jacobi equation using directional derivatives and exposed points, which we think is interesting for its own sake as the associated comparison principle has a very simple proof which avoids all the technical machinery of viscosity solutions.