Continuous-time Markov games with asymmetric information

TL;DR

This paper establishes the existence of the value in a continuous-time Markov game with asymmetric information and provides a simplified formulation of the associated Hamilton-Jacobi equation, enhancing theoretical understanding.

Contribution

It proves the existence of the value in continuous-time asymmetric information games and offers a simpler formulation of the Hamilton-Jacobi equation using directional derivatives.

Findings

Existence of the game value proven using duality techniques.

Unique solution of the Hamilton-Jacobi equation characterized.

Simplified comparison principle for the Hamilton-Jacobi equation.

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. (2015) 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. (2015). 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.