Itô-Wentzell formulas for semimartingale conditional laws with applications to mean-field control

The present paper is an extension of Fadle-Touzi (2024). Following the same methodology, merely based on Taylor expansions, we establish the Itô and Itô-Wentzell formulae for flows of conditional distributions of general semimartingales, thus allowing for discontinuous semimartingales with possibly discontinuous flows of conditional marginals. We apply these results to derive the dynamic programming equations corresponding to mean field control problems with Poisson type common noise and mean field stopping problems with common noise.