Weak Existence and Uniqueness for McKean-Vlasov SDEs with Common Noise

This paper concerns the McKean-Vlasov stochastic differential equation (SDE) with common noise. An appropriate definition of a weak solution to such an equation is developed. The importance of the notion of compatibility in this definition is highlighted by a demonstration of its rôle in connecting...

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Bibliographic Details
Published in:arXiv.org 2020-06
Main Authors: Hammersley, William R P, Šiška, David, Szpruch, Łukasz
Format: Article
Language:English
Subjects:
Coefficient of variation
Dependence
Diffusion coefficient
Linearity
Mathematical analysis
Noise
Partial differential equations
Uniqueness
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Summary:This paper concerns the McKean-Vlasov stochastic differential equation (SDE) with common noise. An appropriate definition of a weak solution to such an equation is developed. The importance of the notion of compatibility in this definition is highlighted by a demonstration of its rôle in connecting weak solutions to McKean-Vlasov SDEs with common noise and solutions to corresponding stochastic partial differential equations (SPDEs). By keeping track of the dependence structure between all components in a sequence of approximating processes, a compactness argument is employed to prove the existence of a weak solution assuming boundedness and joint continuity of the coefficients (allowing for degenerate diffusions). Weak uniqueness is established when the private (idiosyncratic) noise's diffusion coefficient is non-degenerate and the drift is regular in the total variation distance. This seems sharp when one considers using finite-dimensional noise to regularise an infinite dimensional problem. The proof relies on a suitably tailored cost function in the Monge-Kantorovich problem and representation of weak solutions via Girsanov transformations.
ISSN:2331-8422