# Strong Solutions of Mean-Field Stochastic Differential Equations with Irregular Drift

Electronic journal of probability（2018）

Abstract

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 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.

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Key words

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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