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

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 (SDE’s) 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. The worst case measure representing the maximizing player (adversary) is found and it is shown that it is described by an evolution equation on the space of measures on the original state space. N.U. Ahmed is with the School of Information Technology and Engineering and Department of Mathematics University of Ottawa, Ottawa, Ontario, CANADA. E-mail: [email protected] C.D. Charalambous is with the Department of Electrical and Computer Engineering, University of Cyprus, Nicosia, CYPRUS. Also with the School of Information Technology and Engineering, University of Ottawa, Ottawa, Ontario, CANADA. E-mail: [email protected] This work is supported by the European Commission under the project ICCCSYSTEMS and by the University of Cyprus under an internal medium size research project.