It\^o-Wentzell-Lions formulae for flows of full and conditional measures on semimartingales

TL;DR

This paper extends the Itô-Wentzell-Lions formulae to flows of full and conditional measures on general semimartingales, broadening the scope beyond Itô processes with new approximation and localization methods.

Contribution

It introduces generalized Itô-Wentzell-Lions formulae for semimartingales, including special cases involving time-space measures and Poisson-driven functions.

Findings

Established formulae for flows of measures on semimartingales.

Provided approximation techniques for random fields.

Derived specific formulae for Poisson random measures.

Abstract

In this paper, we establish the It\^o-Wentzell-Lions formulae for flows of both full and conditional measures on general semimartingales. This generalizes the existing works on flows of measures on It\^o 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 It\^o-Wentzell-Lions formulae for time-space-measure-dependent functions and for functions driven by Poisson random measures.