Existence and uniqueness theorems for solutions of McKean--Vlasov stochastic equations

TL;DR

This paper establishes new weak and strong existence and uniqueness theorems for multi-dimensional McKean--Vlasov stochastic equations under relaxed regularity conditions, filling gaps in existing literature.

Contribution

It provides the first weak existence results for McKean--Vlasov equations under minimal regularity assumptions, and extends uniqueness results without requiring drift regularity.

Findings

Weak existence under nondegeneracy and linear growth conditions

Weak and strong uniqueness with diffusion depending only on time and state

Analysis based on total variation metric

Abstract

New weak and strong existence and weak and strong uniqueness results for multi-dimensional stochastic McKean--Vlasov equations are established under relaxed regularity conditions. Weak existence is a variation of Krylov's weak existence for It\^o's SDEs under the nondegeneracy of diffusion and no more than a linear growth in the state variable; this part is designed to fill in the existing gap, as earlier such results for McKean-Vlasov equations were not written. Weak and strong uniqueness is established under the restricted assumption of diffusion depending only on time and the state variable, yet without any regularity of the drift in the state variable and also under a linear growth condition on the drift; this part is based on the analysis of the total variation metric.