A risk analysis for a system stabilized by a central agent

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 (Garnier, Papanicolaou and Yang, SIAM J. Fin. Math. 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.

[1]  D. Dawson.  Critical dynamics and fluctuations for a mean-field model of cooperative behavior , 1983 .

[2]  Josselin Garnier,et al.  Large Deviations for a Mean Field Model of Systemic Risk , 2012, SIAM J. Financial Math..

[3]  Mykhaylo Shkolnikov,et al.  Large deviations for interacting Bessel-like processes and applications to systemic risk , 2013, 1303.3061.

[4]  T. Kurtz,et al.  Particle representations for a class of nonlinear SPDEs , 1999 .

[5]  S. Méléard.  Asymptotic behaviour of some interacting particle systems; McKean-Vlasov and Boltzmann models , 1996 .

[6]  Jürgen Gärtner,et al.  Large Deviations, Free Energy Functional and Quasi-Potential for a Mean Field Model of Interacting Diffusions , 1989 .

[7]  X. Zhou,et al.  Stochastic Controls: Hamiltonian Systems and HJB Equations , 1999 .

[8]  A. Sznitman.  Topics in propagation of chaos , 1991 .

[9]  Konstantinos Spiliopoulos,et al.  Default Clustering in Large Pools: Large Deviations , 2013, SIAM J. Financial Math..

[10]  Justin A. Sirignano,et al.  Fluctuation Analysis for the Loss from Default , 2013, 1304.1420.

[11]  Josselin Garnier,et al.  Handbook on Systemic Risk: Diversification in Financial Networks may Increase Systemic Risk , 2013 .

[12]  Hiroshi Tanaka.  Limit Theorems for Certain Diffusion Processes with Interaction , 1984 .

[13]  O. Pironneau,et al.  Dynamic programming for mean-field type control , 2014 .

[14]  P. Dupuis,et al.  Large deviation properties of weakly interacting processes via weak convergence methods , 2010, 1009.6030.

[15]  Jean-Pierre Fouque,et al.  Stability in a Model of Interbank Lending , 2013, SIAM J. Financial Math..

[16]  J. Gärtner,et al.  Large deviations from the mckean-vlasov limit for weakly interacting diffusions , 1987 .

[17]  R. Carmona,et al.  Mean Field Games and Systemic Risk , 2013, 1308.2172.

[18]  Lijun Bo,et al.  Systemic Risk in Interbanking Networks , 2015, SIAM J. Financial Math..

[19]  J. Gärtner.  On the McKean‐Vlasov Limit for Interacting Diffusions , 1988 .

[20]  K. Spiliopoulos,et al.  Default clustering in large portfolios: Typical events. , 2011, 1104.1773.

[21]  Lawrence F. Shampine,et al.  Solving ODEs with MATLAB , 2002 .

[22]  Justin A. Sirignano,et al.  LARGE PORTFOLIO ASYMPTOTICS FOR LOSS FROM DEFAULT , 2011, 1109.1272.

[23]  Amir Dembo,et al.  Large Deviations Techniques and Applications , 1998 .

[24]  W. Runggaldier,et al.  Large portfolio losses: A dynamic contagion model , 2007, 0704.1348.