At the Mercy of the Common Noise: Blow-ups in a Conditional McKean--Vlasov Problem

We extend a model of feedback and contagion in large mean-field systems by introducing a common source of noise driven by Brownian motion. Although the dynamics in the model are continuous, the feedback effect can lead to jump discontinuities in the solutions --- i.e. 'blow-ups'. We prove existence of solutions to the corresponding conditional McKean--Vlasov equation and we show that the pathwise realisation of the common noise can both trigger and prevent blow-ups.