Malliavin differentiability of McKean-Vlasov SDEs with common noise
Jianhai Bao, Goncalo dos Reis, Zac Wilde
TL;DR
This paper addresses Malliavin differentiability for McKean–Vlasov SDEs with common noise under global Lipschitz conditions in both the state and the law. It develops a Malliavin framework by first establishing differentiability for the interacting particle system and then passing to the mean-field limit via propagation of chaos, enabling a meaningful notion of the Malliavin derivative of the conditional law given the common noise. A key contribution is the construction of linear SDEs for the common-noise derivative and the demonstration that these derivatives converge from the IPS to the limit, thereby enabling a conditional integration-by-parts formula on Wiener space for the conditional law $\mu_t$. As an application, the authors derive a conditional integration-by-parts formula that relates Lions derivatives to Skorokhod integrals with respect to the common noise, under a uniform ellipticity condition, which is relevant for probabilistic representations and potential density estimates in mean-field models with common randomness.
Abstract
We establish the Malliavin differentiability of McKean-Vlasov stochastic differential equations (MV-SDEs) with common noise under the global Lipschitz assumption in the space variable and the measure variable. Our result gives also meaning to the Malliavin derivative of the conditional law with respect to the common noise. As an application, we derive an integration by parts formula on the Wiener space for the class of common noise MV-SDEs under consideration.