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Central Limit Results for Jump-Diffusions with Mean Field Interaction and a Common Factor

A system of $N$ weakly interacting particles whose dynamics is given in terms of jump-diffusions with a common factor is considered. The common factor is described through another jump-diffusion and the coefficients of the evolution equation for each particle depend, in addition to its own state value, on the empirical measure of the states of the $N$ particles and the common factor. A Central Limit Theorem, as $N \to \infty$, is established. The limit law is described in terms of a certain Gaussian mixture. An application to models in Mathematical Finance of self-excited correlated defaults is described.