Contributions to the stochastic optimal control of Mckean-Vlasov stochastic différential systems via the derivatives with respect to measures with some applications

Korichi, Fatiha (2024) Contributions to the stochastic optimal control of Mckean-Vlasov stochastic différential systems via the derivatives with respect to measures with some applications. Doctoral thesis, Université Mohamed Khider (Biskra - Algérie).

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Abstract

The central theme is to establish a set of necessary conditions, in the form of stochastic maximum for a di¤erent systems. This thesis is structured around …ve chapters : The …rst 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, Itô formulas, di¤erent methods of solving of optimal control (maximum principle, which has been introduced by Pontryagin et al and dynamical programming principle, wich has been introduced by Bellman) with some di¤erent class of stochastic control, (feedback, singular, impulsional, relaxed, near-optimal, ...etc. In the second chapter, we present the method of the derivative with respect to probability measure. This new approch of derivatives has been introduced by P.L Lions "Cours au Collège de France : Théorie des jeu à champs moyens. http ://www. collegede-france.fr/default/EN/all/equ[1]der/audiovideo.jsp. (2013) » Recently, in the third chapter of this thesis, we study partially observed optimal stochastic singular control problems of general mean-…eld with correlated noises between the system and the observation. The control variable has two components, the …rst being absolutely continuous and the second is a bounded variation, non decreasing continuous on the right with left limits. The dynamic system is governed by Itô-type controlled stochastic differential equation with jumps. The coe¢ cients of the dynamic depend on the state process as well as of its probability law and the continuous control variable.In this work, we formulate this problem mathematically as a combined stochastic continuous control and irregular control problem. We study partially observed optimal stochastic intervention control problem for systems governed by mean-…eld SDEs with correlated noisy between the system and the observation, allowing both classical and intervention control: In the fourth chapter, we establish a second-order stochastic maximum principle for optimal stochastic control of stochastic di¤erential equations of general mean-…eld type. 11Abstract, Korichi. F. 2024 The coe¢ cients of the system are nonlinear and depend on the state process as well as of its probability law. The control variable is allowed to enter into both drift and di¤usion terms. We establish a set of second-order necessary conditions for the optimal control in integral form. The control domain is assumed to be convex. The proof of our main result is based on the the …rst and second-order derivatives with respect to the probability law and by using a convex perturbation with some appropriate estimates. In the …fth chapter, a maximum principle for stochastic model governed by mean-…eld nonlinear controlled Itô-stochastic differential equations is proved. We study the following mean-…eld-type stochastic optimal nonlinear control problem : Minimize a mean-…eld cost functional J (�(�)) = E ZRd �(y�(�); �y�(�))�(dy�); subject to y�(�) solution of the (MF-SDE) : t 2 [0; �] 8>>>>><>>>>>: dy�(t) = RRd ' t; y�(t); �y�(t); �(t)� �(dy�)dt + RRd t; y�(t); �y�(t); �(t)� �(dy�)dW(t); y�(0) = y0: where, �(�) is the control variable valued in a convex bounded subset U � Rk, y� (�) is the controlled state variable, W(�) is a standard Brownian motion, �y�(t) is the distribution of y�(t)

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