Stability, uniqueness and existence of solutions to McKean-Vlasov SDEs: a multidimensional Yamada-Watanabe approach

TL;DR

This paper proves stability, uniqueness, and existence of solutions for multidimensional McKean-Vlasov stochastic differential equations with non-Lipschitz coefficients, using an extended Yamada-Watanabe approach without Lyapunov functions.

Contribution

It extends the Yamada-Watanabe method to multidimensional McKean-Vlasov SDEs, establishing stability, uniqueness, and existence results under minimal regularity assumptions.

Findings

Proved pathwise uniqueness for McKean-Vlasov SDEs with non-Lipschitz coefficients.

Established existence of strong solutions in the deterministic case.

Derived explicit formulas for Lyapunov exponents.

Abstract

We establish stability and pathwise uniqueness of solutions to Wiener noise driven McKean-Vlasov equations with random non-Lipschitz continuous coefficients. In the deterministic case, we also obtain the existence of unique strong solutions. By using our approach, which is based on an extension of the Yamada-Watanabe ansatz to the multidimensional setting and which does not rely on the construction of Lyapunov functions, we prove first moment and pathwise exponential stability. Furthermore, Lyapunov exponents are computed explicitly.