On optimal stochastic control problem of McKean-Vlasov type with some applications via the derivative with respect the law of probability

Abstract

In this thesis, we study the optimal stochastic control for systems governed by McKean- Vlasov stochastic differential equation. of mean-field type. The central theme is the necessary conditions in the form of the Pontryagin’s stochastic maximum of the McKeanVlasov type for optimality with some applications. Recently, the main purpose of this thesis is to derive a set of necessary conditions of optimality, where the differential system is governed by stochastic differential equations of the McKean-Vlasov type. This thesis is structured around three chapters: In the first chapter, we have presented the different class of stochastic control, such as singular controls, relaxed controls, feedback controls, ergodic controls,..etc. . We briefly write the different the well-known methods of solving a stochastic control problem, which are the dynamic programming method and the Pontryagin maximum principle. In the second chapter, we establish the maximum principle for the optimal control for EDS of McKean-Vlasov type.These results have been proved by Andersson D, Djehiche B, See [7]. In the third chapter, we study singular control problem, where control variable is a pair (u(·), ξ(·)) of measurable A1 × A2−valued, Ft−adapted processes, such that ξ(·) is of bounded variation, non-decreasing continuous on the left with right limits and ξ(0−) = 0. Since dξ(t) may be singular with respect to Lebesgue measure dt, we call ξ(·) : the singular part of the control and the process u(·) : its absolutely continuous part. In this chaptre, we established a new set of necessary conditions of optimal singular control, where the system is governed by stochastic differential equations EDSs. In this work, the control domain is not assumed to be convex (i.e., the control domain is a general action space). The derivatives with respect to measure is applied to establish our new result. The results obtained in Chapter 4 are all new and are the subject of a first article entitled: L. Guenane, & M. Hafayed, & S. Meherrem, & S. Abbas: On optimal solutions of general continuous-singular stochastic control problem of McKean-Vlasov type, Journal: Mathematical Methods in the Applied Sciences, Doi.org/10.1002/mma.6392., Volume 43, Issue 10„ Pages 6498-6516 (2020).