Contrôle optimal des systèmes stochastiques partiellement observables

Abstract

The aim of this thesis is to study a stochastic partially observed optimal control problem, for systems of forward backward stochastic di¤erential equations driven by both a family of Teugels martingales and an independent Brownian motion. By using Girsavov’s theorem and a standard spike variational technique, we prove necessary conditions to characterize an optimal control under a partial observation, where the control domain is supposed to be convex. Moreover, under some additional convexity conditions, we prove that these partially observed necessary conditions are su¢ cient. In fact, compared to the existing methods, we get the last achievement in two di¤erent cases according to the linearity or the nonlinearity of the terminal condition for the backward component. As an illustration of the general theory, an application to linear quadratic control problems is also investigated. Noting that this kind of control problems have a powerful tool in the real world of applications. In such problems there is noise in the observation system and the controller is only able to observe partially the state via other variables. For example in …nancial models, one may observe the asset price but not completely its rate of return and/or its volatility, and the portfolio investment in this case is based only on the asset price information. This means that the controller is facing a partial observation control problem.