Mean field stochastic differential equations with a discontinuous diffusion coefficient

TL;DR

This paper investigates mean field stochastic differential equations with discontinuous diffusion coefficients depending on the process's norm, establishing conditions for the existence and uniqueness of solutions and exploring cases where solutions do not exist.

Contribution

It introduces a framework for analyzing mean field SDEs with discontinuous diffusion coefficients and identifies conditions for solution existence and uniqueness.

Findings

Existence and uniqueness of solutions under strong drift.

Identification of cases where global solutions fail.

Analysis of the impact of discontinuous diffusion coefficients.

Abstract

We study $\mathbb{R}^d$-valued mean field stochastic differential equations with a diffusion coefficient depending on the $L_p$-norm of the process in a discontinuous way. We show that under a strong drift there exists a unique global strong solution and consider typical cases where the existence of a global solution fails.