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**Machine Learning and Mean-Field Games**## Optimal bailout strategies resulting from the drift controlled supercooled Stefan problem

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Christa Cuchiero, University of Vienna
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**Wednesday, May 25, 2022**

**Abstract**: We consider a mean-field control 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 to a subset of the entities in order to limit defaults to a given proportion of entities. We prove that the value of the agent's control problem converges as the number of defaultable agents goes to infinity, and that this mean-field limit satisfies a drift controlled version of the supercooled Stefan problem. We compute optimal strategies in feedback form by solving numerically a forward-backward coupled system of PDEs. Our simulations show that the agent's optimal strategy is to subsidize banks whose asset values lie in a non-trivial time-dependent region. We also study a linear-quadratic version of the model where instead of the terminal losses, the agent optimizes a terminal cost function of the equity values. In this case, we are able to give semi-analytic strategies, which we again illustrate numerically. The talk is based on joint work with Christoph Reisinger and Stefan Rigger.