Affiliations: [a] Laboratoire de Probabilités et Modèles Aléatoires & Laboratoire Jacques-Louis Lions, Université Paris Diderot, 75205 Paris Cedex 13, France. E-mail: [email protected] | [b] Department of Mathematics, Stanford University, CA 94305, USA. E-mail: [email protected] | [c] School of Mathematics, University of Minnesota, MN 55455, USA. E-mail: [email protected]
Abstract: We formulate and analyze a multi-agent model for the evolution of individual and systemic risk in which the local agents interact with each other through a central agent who, in turn, is influenced by the mean field of the local agents. The central agent is stabilized by a bistable potential, the only stabilizing force in the system. The local agents derive their stability only from the central agent. In the mean field limit of a large number of local agents we show that the systemic risk decreases when the strength of the interaction of the local agents with the central agent increases. This means that the probability of transition from one of the two stable quasi-equilibria to the other one decreases. We also show that the systemic risk increases when the strength of the interaction of the central agent with the mean field of the local agents increases. Following the financial interpretation of such models and their behavior given in our previous paper (SIAM Journal on Financial Mathematics 4 (2013), 151–184), we may interpret the results of this paper in the following way. From the point of view of systemic risk, and while keeping the perceived risk of the local agents approximately constant, it is better to strengthen the interaction of the local agents with the central agent than the other way around.
Keywords: Mean field models, dynamic phase transitions, systemic risk