Stochastic flocking and its application to systemic risk with jumps

In this dissertation, we first study the effect of jumps on a stochastic flocking model and discuss characteristics of this model. We investigate its application to understand systemic risk by proposing an interbank lending model with jump diffusions and further show that there will be a higher systemic risk with jumps (i.e., sudden increases or decreases in reserves) in our model. Then, to examine how the systemic risk will be affected when each bank is acting toward their best self-interest, we integrate a game feature with jumps where each bank controls its rate of borrowing/lending to a central bank. We then solve Nash equilibria with finitely many players in this game with jumps, within which the central bank acts as a clearing house and adds liquidity to the system. The result indicates that the linear growth contributed by jumps to the system does not affect the systemic risk in the model. Finally, we propose another model with a central bank as well as peripheral banks and investigate the impact of interaction between all banks on systemic risk. The systemic risk might be reduced if the central bank is allowed to monitor liquidity by solving an optimal control problem. We also provide a mean-field game approach to approximate the equilibria for finitely many players.