Distribution dependent SDEs driven by additive continuous noise

TL;DR

This paper investigates distribution dependent stochastic differential equations driven by continuous noise, establishing existence, uniqueness, and convergence results under various relaxed conditions beyond classical Lipschitz assumptions.

Contribution

It extends the theory of distribution dependent SDEs by providing new criteria for well-posedness and convergence under weaker assumptions such as Osgood-continuity and Sobolev differentiability.

Findings

Established well-posedness of distribution dependent SDEs under relaxed conditions.

Proved almost sure convergence of particle systems for various drift regularities.

Extended classical results to broader classes of drifts beyond Lipschitz continuity.

Abstract

We study distribution dependent stochastic differential equation driven by a continuous process, without any specification on its law, following the approach initiated in [16]. We provide several criteria for existence and uniqueness of solutions which go beyond the classical globally Lipschitz setting. In particular we show well-posedness of the equation, as well as almost sure convergence of the associated particle system, for drifts satisfying either Osgood-continuity, monotonicity, local Lipschitz or Sobolev differentiability type assumptions.