Regularization of a mean-field SDE by an additive common noise: The conditional expectation case

TL;DR

This paper studies a McKean-Vlasov SDE with additive common noise and conditional expectation interaction, establishing existence, uniqueness, and propagation of chaos under various noise conditions.

Contribution

It demonstrates that common noise can regularize discontinuous drifts in mean-field SDEs involving conditional expectations, extending existing theory.

Findings

Existence and uniqueness hold with additive individual noise for bounded measurable drifts.

Without individual noise, Lipschitz continuity ensures well-posedness.

Propagation of chaos is established for particle systems with empirical mean interactions.

Abstract

We investigate a McKean-Vlasov stochastic differential equation with an additive common noise and in which the interaction is through the conditional expectation. We show that, in the presence of an additive individual noise, existence and uniqueness of a weak solution hold for any drift given by a bounded and measurable function of the position and the conditional expectation. When there is no individual noise, existence and uniqueness still hold if the drift is in addition Lipschitz in the position variable. This shows that the presence of a finite dimensional common noise may allow to overcome the discontinuity of the drift with respect to the interaction term, provided that this interaction term is a conditional expectation. We also prove propagation of chaos for systems of particles where the conditional expectation is replaced by the empirical mean of the positions or by a closely related contribution with better prepared noise.