Strong Solutions of Mean-Field Stochastic Differential Equations with irregular drift

TL;DR

This paper establishes the existence, uniqueness, and regularity of strong solutions for mean-field stochastic differential equations with irregular drift, extending classical formulas to this setting.

Contribution

It introduces a novel construction method for strong solutions using Malliavin calculus and extends the Bismut-Elworthy-Li formula to mean-field SDEs.

Findings

Proved existence and uniqueness of solutions with irregular drift.

Established Malliavin and Sobolev differentiability of solutions.

Extended Bismut-Elworthy-Li formula to mean-field SDEs.

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.