On the relation between Stratonovich and Ito integrals with functional integrands of conditional measure flows

TL;DR

This paper clarifies the relationship between Ito and Stratonovich integrals in the context of measure-dependent integrands, highlighting the role of Lions derivatives and measure flow correlations in mean-field SDEs.

Contribution

It explicitly derives the correction term connecting Ito and Stratonovich integrals with conditional measure flow integrands, extending understanding in mean-field stochastic calculus.

Findings

Derived explicit correction term involving Lions derivatives

Clarified the relation in the presence of conditional measure flows

Simplified the relation when full measure flows are considered

Abstract

In this small note we explicit the relation between Ito and Stratonovich integrals when conditional measure flow components are present in the integrands. The `correction' term involves Lions-type measure derivatives and clarifies which cross-correlations need to be taken into account. We cast the framework in relation to SDEs of mean-field type depending on conditional flows of measure. The result being trivial under full flows of measure.