Ito's formula for flows of conditional measures on semimartingales

Motivated by recent development of mean-field systems with common noise, this paper establishes Ito's formula for flows of conditional probability measures under a common filtration associated with general semimartingales. This generalizes existing works on flows of conditional measures on Ito processes and flows of deterministic measure on general semimartingales. The key technical components involve constructing conditional independent copies and establishing the equivalence between stochastic integrals with respect to the conditional law of semimartingales and the conditional expectation of stochastic integrals with respect to copies of semimartingales. Ito's formula is then established for cylindrical functions through conditional independent copies, and extended to the general case through function approximations.