Systemic risk governance in a dynamical model of a banking system

We consider the problem of governing systemic risk in a banking system model. The banking system model consists in an initial value problem for a system of stochastic differential equations whose dependent variables are the log-monetary reserves of the banks as functions of time. The banking system model considered generalizes previous models studied in [5], [4], [7] and describes an homogeneous population of banks. Two distinct mechanisms are used to model the cooperation among banks and the cooperation between banks and monetary authority. These mechanisms are regulated respectively by the parameters $\alpha$ and $\gamma$. A bank fails when its log-monetary reserves go below an assigned default level. We call systemic risk or systemic event in a bounded time interval the fact that in that time interval at least a given fraction of the banks fails. The probability of systemic risk in a bounded time interval is evaluated using statistical simulation. A method to govern the probability of systemic risk in a bounded time interval is presented. The goal of the governance is to keep the probability of systemic risk in a bounded time interval between two given thresholds. The governance is based on the choice of the log-monetary reserves of a kind of "ideal bank" as a function of time and on the solution of an optimal control problem for the mean field approximation of the banking system model. The solution of the optimal control problem determines the parameters $\alpha$ and $\gamma$ as functions of time, that is defines the rules of the borrowing and lending activity among banks and between banks and monetary authority. Some numerical examples are discussed. The systemic risk governance is tested in absence and in presence of positive and negative shocks acting on the banking system.