Stability, uniqueness and existence of solutions to McKean-Vlasov SDEs in arbitrary moments

TL;DR

This paper establishes stability, uniqueness, and existence of solutions for McKean-Vlasov stochastic differential equations with non-Lipschitz drifts and random coefficients, providing new moment estimates and stability results.

Contribution

It introduces a novel analysis for McKean-Vlasov SDEs with non-Lipschitz drifts, including moment estimates and stability results that do not rely on the existence of Lyapunov functions.

Findings

Proves stability and pathwise uniqueness for McKean-Vlasov SDEs with random coefficients.

Provides moment estimates for non-Lipschitz drift coefficients.

Establishes p-th moment and exponential stability with explicit Lyapunov exponents.

Abstract

We deduce stability and pathwise uniqueness for a McKean-Vlasov equation with random coefficients and a multidimensional Brownian motion as driver. Our analysis focuses on a non-Lipschitz drift coefficient and includes moment estimates for random It\^o processes that are of independent interest. For deterministic coefficients we provide unique strong solutions, even if the drift fails to be of affine growth. The theory that we develop rests on It\^o's formula and leads to $p$-th moment and pathwise $\alpha$-exponential stability for $p\geq 2$ and $\alpha > 0$ with explicit Lyapunov exponents, regardless of whether a Lyapunov function exists.