Stochastic nonlinear minimax dynamic games with noisy measurements

The paper is concerned with nonlinear stochastic minimax dynamic games which are subject to noisy measurements. The minimizing players are control inputs while the maximizing players are square-integrable stochastic processes. First, the minimax dynamic game is formulated using an information state, which satisfies a stochastic partial differential equation. Subsequently, a separation theorem is derived between the estimation and the control problems. Second, a certainty-equivalence principle is introduced along the lines of Whittle (1991), by defining the future stress, the past stress, the minimum stress and the certainty-equivalence controller. Third, the separation theorem and the certainty-equivalence principle are applied to solve the linear-quadratic-Gaussian minimax game. The optimal control and the certainty-equivalence control are shown to be identical. The results of the paper generalize the L²-gain of deterministic systems to stochastic analogs; they are related to the controller design of stochastic risk-sensitive control and minimax deterministic dynamic games