  We consider the problem faced by a central bank which bails out distressed
financial institutions that pose systemic risk to the banking sector. In a
structural default model with mutual obligations, the central agent seeks to
inject a minimum amount of cash in order to limit defaults to a given
proportion of entities. We prove that the value of the central agent's control
problem converges as the number of defaultable institutions goes to infinity,
and that it satisfies a drift controlled version of the supercooled Stefan
problem. We compute optimal strategies in feedback form by solving numerically
a regularized version of the corresponding mean field control problem using a
policy gradient method. Our simulations show that the central agent's optimal
strategy is to subsidise banks whose equity values lie in a non-trivial
time-dependent region.
