Before revision:  Although the dynamics in the model are continuous, the  feedback effect can lead to jump discontinuities in the solutions, which we refer to as `blow-ups' . We prove existence of solutions to the corresponding conditional McKean--Vlasov equation, by realising them as suitable limit points of the finite-dimensional particle system , and we show that the pathwise realisation of the common noise can both trigger and prevent blow-ups .
After revision:  Although the dynamics in the model are continuous, the feedback effect can lead to `blow-up' phenomena whereby solutions develop jump-discontinuities. Our main results are twofold and concern the conditional McKean--Vlasov equation, by realising them as suitable limit points of the finite-dimensional particle system , and we show that the pathwise realisation of the common noise can both trigger and prevent blow-ups .