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

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 Itô 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 Itô's formula and leads to $p$-th moment and pathwise $\alpha$-exponential stability for $p\ge 2$ and $\alpha >0$ with explicit Lyapunov exponents, regardless of whether a Lyapunov function exists.

References in corpus (2)

• Mean-field backward stochastic differential equations: A limit approach
• Stability, uniqueness and existence of solutions to McKean-Vlasov SDEs: a multidimensional Yamada-Watanabe approach