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

In this small note we explicit the relation between Itô 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.

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