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

William R. P. Hammersley, David Šiška, Łukasz Szpruch

Published 2019-08-02Version 1

This paper concerns the McKean-Vlasov stochastic differential equation (SDE) with common noise, a distribution dependent SDE with a conditional non-linearity. Such equations describe the limiting behaviour of a representative particle in a mean-field interacting system driven by correlated noises as the particle number tends to infinity. 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 r\^ole 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 for the approximation process, 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 noise's diffusion coefficient is non-degenerate and the drift is regular in the total variation distance. This seems sharp when one considers finite-dimensional noise. The proof relies on a suitably tailored cost function in the Monge-Kantorovich problem and extends a remarkable technique based on Girsanov transformations previously employed in the case of uncorrelated noises to the common noise setting.