An Inverse Problem for Adaptive Linear Quadratic Stochastic Differential Games

In this paper, we consider an inverse problem for adaptive two-player stochastic linear quadratic differential games where the cost functions of players are unknown to each other, which arise in many practical situations but have rarely been explored theoretically. We will introduce a new feedback pattern where Player 2 chooses the optimal actions given the actions of Player 1, while Player 1 uses an adaptive learning algorithm to recover Player 2’s cost function and chooses an adaptive strategy by solving the corresponding coupled Riccati equations. Under some suitable assumptions on the system matrices, it is shown that estimates for the cost function parameters are strongly consistent, and the adaptive game systems asymptotically reach the Nash equilibrium.