Strong well-posedness of McKean-Vlasov stochastic differential equation with H{ö}lder drift

Abstract

In this paper, we prove pathwise uniqueness for stochastic systems of McKean-Vlasov type with singular drift, even in the measure argument, and uniformly non-degenerate Lipschitz diffusion matrix. Our proof is based on Zvonkin's transformation \cite{zvonkin\_transformation\_1974} and so on the regularization properties of the associated PDE, which is stated on the space $[0,T]\times {\mathbb{R}}^d\times \mathcal{P}\_2({\mathbb{R}}^d)$, where $T$ is a positive number, $d$ denotes the dimension equation and $\mathcal{P}\_2({\mathbb{R}}^d)$ is the space of probability measures on ${\mathbb{R}}^d$ with finite second order moment. Especially, a smoothing effect in the measure direction is exhibited. Our approach is based on a parametrix expansion of the transition density of the McKean-Vlasov process.

References in corpus (1)

• Mean field games with common noise

Cited by in corpus (4)

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• On Explicit Milstein-type Scheme for Mckean-Vlasov Stochastic Differential Equations with Super-linear Drift Coefficient
• Weak Solutions to Vlasov-McKean Equations under Lyapunov-Type Conditions
• Approximations of Mckean-Vlasov SDEs with Irregular Coefficients