Stability, Uniqueness and Existence of Solutions to McKean–Vlasov Stochastic Differential Equations in Arbitrary Moments

Alexander Kalinin (), Thilo Meyer-Brandis () and Frank Proske ()
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Alexander Kalinin: LMU Munich
Thilo Meyer-Brandis: LMU Munich
Frank Proske: University of Oslo

Journal of Theoretical Probability, 2024, vol. 37, issue 4, 2941-2989

Abstract: 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 continuous drift 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 pth moment and pathwise exponential stability for $$p\ge 2$$ p ≥ 2 with explicit Lyapunov exponents.

Keywords: McKean–Vlasov equation; Lyapunov stability; Moment estimate; Asymptotic behaviour; Pathwise uniqueness; Strong solution; Non-Lipschitz drift; Itô process; 60H20; 60H30; 60F25; 37H30; 45M10 (search for similar items in EconPapers)
Date: 2024
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DOI: 10.1007/s10959-024-01344-2

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