Minimax games for stochastic systems subject to relative entropy uncertainty: Applications to SDEs on Hilbert spaces

Abstract

In this paper, we consider minimax games for stochastic uncertain systems with the pay-off being a nonlinear functional of the uncertain measure where the uncertainty is measured in terms of relative entropy between the uncertain and the nominal measure. The maximizing player is the uncertain measure, while the minimizer is the control which induces a nominal measure. Existence and uniqueness of minimax solutions are derived on suitable spaces of measures. Several examples are presented illustrating the results. Subsequently, the results are also applied to controlled stochastic differential equations on Hilbert spaces. Based on infinite dimensional extension of Girsanov's measure transformation, martingale solutions are used in establishing existence and uniqueness of minimax strategies. Moreover, some basic properties of the relative entropy of measures on infinite dimensional spaces are presented and then applied to uncertain systems described by a stochastic differential inclusion on Hilbert space. An explicit expression for the worst case measure representing the maximizing player (adversary) is found. © Springer-Verlag London Limited 2006.