The relaxed optimal control problem for Mean-Field SDEs systems and application

Chala, Adel

doi:10.1016/j.automatica.2013.12.022

Cited by 18 publications

(16 citation statements)

References 13 publications

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“…For general McKean-Vlasov controlled equations, such a limit theory has been proved in Lacker [51] in a context without common noise, where an essential tool is a compactness argument, which is made accessible by formulating an appropriate relaxed control for McKean-Vlasov equations, in the spirit of El Karoui, Huu Nguyen, and Jeanblanc-Picqué [27], and by introducing suitable martingale problems, similar to those of Stroock and Varadhan [74]. The same formulation and arguments have also been used in Bahlali, Mezerdi, and Mezerdi [4; 5; 6; 7] and Chala [22] to study stability and approximation problems.…”

Section: Introductionmentioning

confidence: 99%

McKean-Vlasov optimal control: limit theory and equivalence between different formulations

Djete,

Possamaï,

Tan

2020

Preprint

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We study a McKean-Vlasov optimal control problem with common noise, in order to establish the corresponding limit theory, as well as the equivalence between different formulations, including the strong, weak and relaxed formulation. In contrast to the strong formulation, where the problem is formulated on a fixed probability space equipped with two Brownian filtrations, the weak formulation is obtained by considering a more general probability space with two filtrations satisfying an (H)-hypothesis type condition from the theory of enlargement of filtrations. When the common noise is uncontrolled, our relaxed formulation is obtained by considering a suitable controlled martingale problem. As for classical optimal control problems, we prove that the set of all relaxed controls is the closure of the set of all strong controls, when considered as probability measures on the canonical space. Consequently, we obtain the equivalence of the different formulations of the control problem, under additional mild regularity conditions on the reward functions. This is also a crucial technical step to prove the limit theory of the McKean-Vlasov control problem, that is to say proving that it consists in the limit of a large population control problem with common noise.

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Section: Introductionmentioning

confidence: 99%

McKean-Vlasov optimal control: limit theory and equivalence between different formulations

Djete,

Possamaï,

Tan

2020

Preprint

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“…So it is not until Buckdahn et al [4,5] established the theory of the mean-field BSDEs that the stochastic maximum principle for the optimal control system of mean-field type has become a popular topic. Interested readers may refer to [1][2][3]6,7,9,12,13,16,22,23,28,29,31,32,34] for various versions of the stochastic maximum principles for the mean-field models.…”

Section: Introductionmentioning

confidence: 99%

Linear‐Quadratic Optimal Control Problems for Mean‐Field Stochastic Differential Equations with Jumps

Tang

Meng

2018

Asian Journal of Control

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In this paper, we study a linear-quadratic optimal control problem for mean-field stochastic differential equations driven by a Poisson random martingale measure and a one-dimensional Brownian motion. Firstly, the existence and uniqueness of the optimal control is obtained by the classic convex variation principle. Secondly, by the duality method, the optimality system, also called the stochastic Hamilton system which turns out to be a linear fully coupled mean-field forward-backward stochastic differential equation with jumps, is derived to characterize the optimal control. Thirdly, applying a decoupling technique, we establish the connection between two Riccati equations and the stochastic Hamilton system and then prove the optimal control has a state feedback representation.

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“…The stochastic maximum principle approach was also used in a context involving conditional distributions in Buckdahn, Li, and Ma [20] and Carmona and Zhu [26]. Related results have been obtained in the context of a relaxed formulation of the control problem, allowing in addition to obtain existence results by Chala [29], which were then revisited by Lacker [52], and Bahlali, Mezerdi, and Mezerdi [4; 5; 6] Readers familiar with the classical theory of stochastic control know that another popular approach to the problem is to use Bellman's optimality principle, to obtain the so-called dynamic programming principle. In a nutshell, the idea behind the DPP is that the global optimisation problem can be solved by a recursive resolution of successive local optimisation problems.…”

Section: The Pioneering Work On Mckean-vlasov Equations Is Due To Mckeanmentioning

confidence: 99%

McKean-Vlasov optimal control: the dynamic programming principle

Djete¹,

Possamaï²,

Tan³

2019

Preprint

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We study the McKean-Vlasov optimal control problem with common noise in various formulations, namely the strong and weak formulation, as well as the Markovian and non-Markovian formulations, and allowing for the law of the control process to appear in the state dynamics. By interpreting the controls as probability measures on an appropriate canonical space with two filtrations, we then develop the classical measurable selection, conditioning and concatenation arguments in this new context, and establish the dynamic programming principle under general conditions. * Mao Fabrice Djete gratefully acknowledges support from the région Île-de-France. This work also benefited from support of the ANR project PACMAN ANR-16-CE05-0027.

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The relaxed optimal control problem for Mean-Field SDEs systems and application

Cited by 18 publications

References 13 publications

McKean-Vlasov optimal control: limit theory and equivalence between different formulations

McKean-Vlasov optimal control: limit theory and equivalence between different formulations

Linear‐Quadratic Optimal Control Problems for Mean‐Field Stochastic Differential Equations with Jumps

McKean-Vlasov optimal control: the dynamic programming principle

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