Sur les conditions nécessaires et suffisantes d’optimalité pour une classe des contrôles stochastiques mixed de type champ-moyen et leurs applications aux finances.

Abstract

This thesis is concerned with stochastic control of mean-field type. The central theme isthe necessary and sufficient conditions in the form of the Pontryagin’s stochastic maximum of the mean-field type for optimality with partial information and some applications. Recently, the main purpose of this thesis is to derive a set of necessary as well as sufficient conditions of optimality with partial information, where the system is governed by stochastic differential equations of the mean field type. This thesis is structured around three chapters : The first chapter is essentially a reminder. we presents some concepts and results that allow us to prove our results, such as stochastic processes, conditional expectation, martingales, Ito formulas, class of stochastic control, etc. In the second chapter, we have proved the necessary and sufficient conditions of near-optimality of order "λ satisfied by an optimal stochastic control, where the system is governed by stochastic differential equations EDSs. The stochastic control domain is assumed to be convex. The method used is based on the Ekeland lemma. The results obtained in Chapter 2 are all new and are the subject of a first article entitled: Boukaf Samira & Mokhtar Hafayed and Ghebouli Messaoud : A study on optimal control problem with "λ-error bound for stochastic systems with application to linear quadratic problem, International Journal of Dynamics and Control, Springer DOI : 10.1007 / s40435- 015-0178-x (2017), Volume 5, Issue 2, pp 297–305 (2017) In the third chapter, we have proved the stochastic maximum principle under partial information, where the system is governed by forward backward stochastic differential equations (FBSDEs) deriven by Lévy process. These results have been applied to solve an optimization problem in finance. Moreover, as an application, we study a partial information mean-variance portfolio selection problem, driven by Teugels martingales associated with Gamma process, where the explicit optimal portfolio strategy is derived in feedback form. The results obtained in Chapter 3 are all new and are the subject of a second article entitled : Mokhtar Hafayed, & Ghebouli Messaoud & Samira Boukaf & Yan Shi : Partial information optimal control of mean-field forward-backward stochastic system driven by Teugels martingales with applications (2016) DOI 10.1016/j.neucom. 2016.03.002. Neurocomputing, Vol 200 pages 11–21 (2016)