  We consider a mean-field model for large banking systems, which takes into
account default and recovery of the institutions. Building on models used for
groups of interacting neurons, we first study a McKean-Vlasov dynamics and its
evolutionary Fokker-Planck equation in which the mean-field interactions occur
through a mean-reverting term and through a hitting time corresponding to a
default level. The latter feature reflects the impact of a financial
institution's default on the global distribution of reserves in the banking
system. The systemic risk problem of financial institutions is understood as a
blow-up phenomenon of the Fokker-Planck equation. Then, we incorporate in the
model an optimization component by letting the institutions control part of
their dynamics in order to minimize their expected risk. Phrasing this
optimization problem as a mean-field game, we provide an explicit solution in a
special case and, in the general case, we report numerical experiments based on
a finite difference scheme.
