Electronic Journal of Probability

Strong solutions of mean-field stochastic differential equations with irregular drift

Martin Bauer, Thilo Meyer-Brandis, and Frank Proske

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We investigate existence and uniqueness of strong solutions of mean-field stochastic differential equations with irregular drift coefficients. Our direct construction of strong solutions is mainly based on a compactness criterion employing Malliavin Calculus together with some local time calculus. Furthermore, we establish regularity properties of the solutions such as Malliavin differentiablility as well as Sobolev differentiability and Hölder continuity in the initial condition. Using this properties we formulate an extension of the Bismut-Elworthy-Li formula to mean-field stochastic differential equations to get a probabilistic representation of the first order derivative of an expectation functional with respect to the initial condition.

Article information

Electron. J. Probab., Volume 23 (2018), paper no. 132, 35 pp.

Received: 5 July 2018
Accepted: 13 December 2018
First available in Project Euclid: 22 December 2018

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Zentralblatt MATH identifier

Primary: 60H07: Stochastic calculus of variations and the Malliavin calculus 60H10: Stochastic ordinary differential equations [See also 34F05] 60H30: Applications of stochastic analysis (to PDE, etc.) 60H20: Stochastic integral equations

mean-field stochastic differential equation McKean-Vlasov equation strong solutions irregular coefficients Malliavin calculus local-time integral Sobolev differentiability in the initial condition Bismut-Elworthy-Li formula

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Bauer, Martin; Meyer-Brandis, Thilo; Proske, Frank. Strong solutions of mean-field stochastic differential equations with irregular drift. Electron. J. Probab. 23 (2018), paper no. 132, 35 pp. doi:10.1214/18-EJP259. https://projecteuclid.org/euclid.ejp/1545447917

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