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

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 de...

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Bibliographic Details
Published in:Journal of theoretical probability 2024-11, Vol.37 (4), p.2941-2989
Main Authors: Kalinin, Alexander, Meyer-Brandis, Thilo, Proske, Frank
Format: Article
Language:English
Subjects:
Differential equations
Drift estimation
Liapunov exponents
Mathematics
Mathematics and Statistics
Motion stability
Probability Theory and Stochastic Processes
Statistics
Uniqueness
Vlasov equations
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Summary: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 p th moment and pathwise exponential stability for p ≥ 2 with explicit Lyapunov exponents.
ISSN:0894-9840
1572-9230
DOI:10.1007/s10959-024-01344-2