Stability of McKean–Vlasov stochastic differential equations and applications

Autor: Brahim Mezerdi, Khaled Bahlali, Mohamed Amine Mezerdi
Přispěvatelé: Institut de Mathématiques de Toulon - EA 2134 (IMATH), Université de Toulon (UTLN), Université Mohamed Khider de Biskra (BISKRA)
Jazyk: angličtina
Rok vydání: 2019
Předmět:
Zdroj: Stochastics and Dynamics
Stochastics and Dynamics, World Scientific Publishing, 2019, 20 (01), pp.2050007. ⟨10.1142/S0219493720500070⟩
ISSN: 0219-4937
DOI: 10.1142/S0219493720500070⟩
Popis: International audience; We consider McKean–Vlasov stochastic differential equations (MVSDEs), which are SDEs where the drift and diffusion coefficients depend not only on the state of the unknown process but also on its probability distribution. This type of SDEs was studied in statistical physics and represents the natural setting for stochastic mean-field games. We will first discuss questions of existence and uniqueness of solutions under an Osgood type condition improving the well-known Lipschitz case. Then, we derive various stability properties with respect to initial data, coefficients and driving processes, generalizing known results for classical SDEs. Finally, we establish a result on the approximation of the solution of a MVSDE associated to a relaxed control by the solutions of the same equation associated to strict controls. As a consequence, we show that the relaxed and strict control problems have the same value function. This last property improves known results proved for a special class of MVSDEs, where the dependence on the distribution was made via a linear functional.
Databáze: OpenAIRE
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