Two-Player Diffusion Control Games with Private Information

Abstract

This paper presents a two-player stochastic differential game with diffusion control and rewards that are zero-sum. The players have i.i.d. exponentially distributed time horizons at which their states are compared and only the best player receives a reward. We assume that players have no information about their opponent and can only use private information for controlling their state. We show that there always exists a Markovian saddle point. Moreover, we consider the existence of saddle points in the class of threshold controls, i.e., controls choosing the maximal volatility below some threshold and the minimal above. There exists a symmetric saddle point of threshold type in closed form if and only if the ratio of maximal and minimal volatility does not exceed the value \(\sqrt{2}\).