Itô-Wentzell-Lions formulae for flows of full and conditional measures on semimartingales
Liu Jisheng, Zhang Jing
TL;DR
This work develops Itô-Wentzell-Lions formulae for flows of both full and conditional probability measures driven by general semimartingales, unifying diffusion and jump settings within a mean-field framework. The core method combines a three-stage approximation via cylindrical functions with localization to preserve adaptiveness, enabling passage from cylindrical to general RF-Partially-$ ext{C}^2$ functionals. The authors provide explicit expansions for two key specializations: time-space-measure dependent functions and Poisson-driven systems, thereby broadening applicability to McKean-Vlasov control, mean-field games, and common-noise models. The results offer a rigorous analytic toolkit for analyzing stochastic systems where coefficients depend on the distribution of the state, with implications for both theoretical study and numerical approximation in mean-field problems.
Abstract
In this paper, we establish the Itô-Wentzell-Lions formulae for flows of both full and conditional measures on general semimartingales. This generalizes the existing works on flows of measures on Itô processes. The key technical components involve an appropriate approximation of random fields by cylindrical functions and localization techniques. Moreover, we present the specific formulae in two special cases, including Itô-Wentzell-Lions formulae for time-space-measure-dependent functions and for functions driven by Poisson random measures.