Partially obseved optimal control problem for SDEs of Mckean-Vlasov type and Applications

Abstract

Partially observed control problems have received much attention and became a powerful tool in many fields, such as mathematical finance, optimal control, etc.From the viewpoint of reality, many situations, full information is not always available to controllers, but the partial one with noise. Furthermore, the recent work of Buckdahn, R. [7] and Hafayed, M. [24] on Mckean-Vlasov type stochastic differential equations and their optimal control opens a new avenue for the study of optimal control problems in general. The objective of this thesis is to extend these results of [7] and [24] to the case of a partially observed optimal control problem. More precisely, we study partially observed optimal control problems of general McKean-Vlasov differential equations, in which the coefficients depend on the state of the solution process as well as of its law and the control variable. By applying Girsanov’s theorem with a standard convex variational technique, we develop the stochastic maximum principle for our partially observed control problem where the control domain is convex. Also, in this thesis, we prove a new stochastic maximum principle for a class of partially observed optimal control problems of Mckean-Vlasov type with jumps. The stochastic system under consideration is governed by a stochastic differential equation driven by Poisson random measure and an independent Brownian motion. The derivatives with respect to probability measure and the associate Itô-formula are applied to prove our main results. And as an illustration, by applying our maximum principle, McKean-Vlasov type linear quadratic control problem with jump is discussed,where the partially observed optimal control is obtained explicitly in feedback form.Key words. Partially observed optimal control, Stochastic maximum principle, Derivatives with respect to the measure, McKean-Vlasov differential equations, McKean-Vlasov stochastic system with jumps, Probability measure, Girsanov’s theorem