Stochastic Adaptive Linear Quadratic Differential Games

Game theory is playing more and more important roles in understanding complex systems and in investigating intelligent machines with various uncertainties. As a starting point, we consider the classical two-player zero-sum linear-quadratic stochastic differential games, but in contrast to most of the existing studies, the coefficient matrices of the systems are assumed to be unknown to both players, and consequently it is necessary to study adaptive strategies of the players, which may be termed as adaptive games and which has rarely been explored in the literature. In this paper, it will be shown that the adaptive strategies of both players can be constructed by the combined use of a weighted least squares (WLS) estimation algorithm, a random regularization method and a diminishing excitation method. Under almost the same structural conditions as those in the traditional known parameters case, we will show that the closed-loop adaptive game systems will be globally stable and asymptotically reaches the Nash equilibrium as the time tends to infinity.