Large Deviations for a Mean Field Model of Systemic Risk

We consider a system of diffusion processes that interact through their empirical mean and have a stabilizing force acting on each of them, corresponding to a bistable potential. There are three parameters that characterize the system: the strength of the intrinsic stabilization, the strength of the external random perturbations, and the degree of cooperation or interaction between them. The last one is the rate of mean reversion of each component to the empirical mean of the system. We interpret this model in the context of systemic risk and analyze in detail the effect of cooperation between the components, that is, the rate of mean reversion. We show that in a certain regime of parameters increasing cooperation tends to increase the stability of the individual agents, but it also increases the overall or systemic risk. We use the theory of large deviations of diffusions interacting through their mean field.

[1]  P. Moral,et al.  Genealogical particle analysis of rare events , 2005, math/0602525.

[2]  Ofer Zeitouni,et al.  Increasing propagation of chaos for mean field models , 1999 .

[3]  M. Freidlin,et al.  Random Perturbations of Dynamical Systems , 1984 .

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

[5]  Stefano Battiston,et al.  Systemic risk in a unifying framework for cascading processes on networks , 2009, 0907.5325.

[6]  D. Dawson,et al.  Multilevel large deviations and interacting diffusions , 1994 .

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

[8]  David G. Rand,et al.  Individual versus systemic risk and the Regulator's Dilemma , 2011, Proceedings of the National Academy of Sciences.

[9]  G. B. Arous,et al.  Large deviations for Langevin spin glass dynamics , 1995 .

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

[11]  S. Varadhan,et al.  Large deviations , 2019, Graduate Studies in Mathematics.

[12]  J. Yang,et al.  Network Models and Financial Stability , 2008 .

[13]  R. May,et al.  Systemic risk: the dynamics of model banking systems , 2010, Journal of The Royal Society Interface.

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

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

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

[17]  H. Haken Synergetics: an Introduction, Nonequilibrium Phase Transitions and Self-organization in Physics, Chemistry, and Biology , 1977 .

[18]  J. Stiglitz Risk and Global Economic Architecture: Why Full Financial Integration May Be Undesirable , 2010 .

[19]  Hermann Haken,et al.  Synergetics: An Introduction , 1983 .

[20]  Peter Imkeller,et al.  Large deviations and a Kramers’ type law for self-stabilizing diffusions , 2006 .

[21]  P. Moral,et al.  Concentration Inequalities for Mean Field Particle Models , 2011, 1211.1837.

[22]  Prasanna Gai,et al.  Contagion in financial networks , 2010, Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences.

[23]  P. Dupuis,et al.  Large deviations for infinite dimensional stochastic dynamical systems , 2008, 0808.3631.

[24]  S. Battiston,et al.  Liaisons Dangereuses: Increasing Connectivity, Risk Sharing, and Systemic Risk , 2009 .

[25]  Pierre Del Moral,et al.  Large Deviations for Interacting Processes in the Strong Topology , 2005 .

[26]  P. Moral,et al.  Large deviations for interacting particle systems: Applications to non-linear filtering , 1998 .

[27]  S. Kapadia,et al.  Andrew G Haldane: Rethinking the Financial Network , 2022 .

[28]  Johan Walden,et al.  Diversification disasters , 2010 .

[29]  Stefano Battiston,et al.  Default Cascades: When Does Risk Diversification Increase Stability? , 2011 .

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

[31]  G. B. Arous,et al.  Large deviations for Langevin spin glass dynamics , 1995 .

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

[33]  Alan G. White,et al.  Valuing Credit Default Swaps II , 2000 .

[34]  Li-Hsien Sun Systemic Risk Illustrated , 2014 .

[35]  A. Lo,et al.  A Survey of Systemic Risk Analytics , 2012 .

[36]  R. May,et al.  Systemic risk in banking ecosystems , 2011, Nature.