We consider a dynamic model of interconnected banks. New banks can emerge,
and existing banks can default, creating a birth-and-death setup.
Microscopically, banks evolve as independent geometric Brownian motions.
Systemic effects are captured through default contagion: as one bank defaults,
reserves of other banks are reduced by a random proportion. After examining the
long-term stability of this system, we investigate mean-field limits as the
number of banks tends to infinity. Our main results concern the measure-valued
scaling limit which is governed by a McKean-Vlasov jump-diffusion. The default
impact creates a mean-field drift, while the births and defaults introduce jump
terms tied to the current distribution of the process. Individual dynamics in
the limit is described by the propagation of chaos phenomenon. In certain
cases, we explicitly characterize the limiting average reserves.