The Annals of Applied Probability

Particle systems with singular interaction through hitting times: Application in systemic risk modeling

Sergey Nadtochiy and Mykhaylo Shkolnikov

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We propose an interacting particle system to model the evolution of a system of banks with mutual exposures. In this model, a bank defaults when its normalized asset value hits a lower threshold, and its default causes instantaneous losses to other banks, possibly triggering a cascade of defaults. The strength of this interaction is determined by the level of the so-called noncore exposure. We show that, when the size of the system becomes large, the cumulative loss process of a bank resulting from the defaults of other banks exhibits discontinuities. These discontinuities are naturally interpreted as systemic events, and we characterize them explicitly in terms of the level of noncore exposure and the fraction of banks that are “about to default.” The main mathematical challenges of our work stem from the very singular nature of the interaction between the particles, which is inherited by the limiting system. A similar particle system is analyzed in [Ann. Appl. Probab. 25 (2015) 2096–2133] and [Stochastic Process. Appl. 125 (2015) 2451–2492], and we build on and extend their results. In particular, we characterize the large-population limit of the system and analyze the jump times, the regularity between jumps, and the local uniqueness of the limiting process.

Article information

Ann. Appl. Probab., Volume 29, Number 1 (2019), 89-129.

Received: May 2017
Revised: November 2017
First available in Project Euclid: 5 December 2018

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Zentralblatt MATH identifier

Primary: 35B65: Smoothness and regularity of solutions 35K20: Initial-boundary value problems for second-order parabolic equations 82C22: Interacting particle systems [See also 60K35] 91G80: Financial applications of other theories (stochastic control, calculus of variations, PDE, SPDE, dynamical systems)

Banking systems blow-ups in parabolic partial differential equations default cascades interacting particle systems large system limits loss of continuity mean-field models noncore exposures nonlinear Cauchy–Dirichlet problems regularity estimates self-excitation singular interaction systemic crises systemic risk


Nadtochiy, Sergey; Shkolnikov, Mykhaylo. Particle systems with singular interaction through hitting times: Application in systemic risk modeling. Ann. Appl. Probab. 29 (2019), no. 1, 89--129. doi:10.1214/18-AAP1403.

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  • [1] Alichi, A., Ryoo, S. C. and Hong, C. (2012). Managing non-core liabilities and leverage of the banking system: A building block for macroprudential policy making in Korea. Technical report, International Monetary Fund, Washington, DC.
  • [2] Battiston, S., Gatti, D. D., Gallegati, M., Greenwald, B. C. and Stiglitz, J. E. (2009). Liaisons dangereuses: Increasing connectivity, risk sharing, and systemic risk. Technical report, National Bureau of Economic Research, Cambridge, MA.
  • [3] Bush, N., Hambly, B. M., Haworth, H., Jin, L. and Reisinger, C. (2011). Stochastic evolution equations in portfolio credit modelling. SIAM J. Financial Math. 2 627–664.
  • [4] Cáceres, M. J., Carrillo, J. A. and Perthame, B. (2011). Analysis of nonlinear noisy integrate & fire neuron models: Blow-up and steady states. J. Math. Neurosci. 1 Art. 7, 33.
  • [5] Cáceres, M. J. and Perthame, B. (2014). Beyond blow-up in excitatory integrate and fire neuronal networks: Refractory period and spontaneous activity. J. Theoret. Biol. 350 81–89.
  • [6] Carmona, R., Fouque, J.-P. and Sun, L.-H. (2015). Mean field games and systemic risk. Commun. Math. Sci. 13 911–933.
  • [7] Carrillo, J. A., Perthame, B., Salort, D. and Smets, D. (2015). Qualitative properties of solutions for the noisy integrate and fire model in computational neuroscience. Nonlinearity 28 3365–3388.
  • [8] Dai Pra, P., Runggaldier, W. J., Sartori, E. and Tolotti, M. (2009). Large portfolio losses: A dynamic contagion model. Ann. Appl. Probab. 19 347–394.
  • [9] Delarue, F., Inglis, J., Rubenthaler, S. and Tanré, E. (2015). Global solvability of a networked integrate-and-fire model of McKean–Vlasov type. Ann. Appl. Probab. 25 2096–2133.
  • [10] Delarue, F., Inglis, J., Rubenthaler, S. and Tanré, E. (2015). Particle systems with a singular mean-field self-excitation. Application to neuronal networks. Stochastic Process. Appl. 125 2451–2492.
  • [11] Dembo, A., Deuschel, J.-D. and Duffie, D. (2004). Large portfolio losses. Finance Stoch. 8 3–16.
  • [12] Dembo, A. and Tsai, L.-C. (2017). Criticality of a randomly-driven front. ArXiv:1705.10017.
  • [13] Duffie, D. and Zhu, H. (2011). Does a central clearing counterparty reduce counterparty risk? Rev. Asset Pricing Stud. 1 74–95.
  • [14] Gai, P. and Kapadia, S. (2010). Contagion in financial networks. Technical report, Bank of England, London.
  • [15] Giesecke, K., Spiliopoulos, K., Sowers, R. B. and Sirignano, J. A. (2015). Large portfolio asymptotics for loss from default. Math. Finance 25 77–114.
  • [16] Glasserman, P. (2015). Contagion in financial networks. Technical report, Office of Financial Research.
  • [17] Horst, U. (2007). Stochastic cascades, credit contagion, and large portfolio losses. J. Econ. Behav. Organ. 63 25–54.
  • [18] Inglis, J. and Talay, D. (2015). Mean-field limit of a stochastic particle system smoothly interacting through threshold hitting-times and applications to neural networks with dendritic component. SIAM J. Math. Anal. 47 3884–3916.
  • [19] Jacod, J. and Shiryaev, A. N. (2003). Limit Theorems for Stochastic Processes, 2nd ed. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences] 288. Springer, Berlin.
  • [20] Kesten, H. and Sidoravicius, V. (2008). A problem in one-dimensional diffusion-limited aggregation (DLA) and positive recurrence of Markov chains. Ann. Probab. 36 1838–1879.
  • [21] Krylov, N. V. (2009). Controlled Diffusion Processes. Stochastic Modelling and Applied Probability 14. Springer, Berlin. Translated from the 1977 Russian original by A. B. Aries, Reprint of the 1980 edition.
  • [22] Ladyženskaya, O. A. (1954). On solvability of the fundamental boundary problems for equations of parabolic and hyperbolic type. Dokl. Akad. Nauk SSSR 97 395–398.
  • [23] Ladyženskaya, O. A., Solonnikov, V. A. and Ural’ceva, N. N. (1968). Linear and Quasilinear Equations of Parabolic Type. Translations of Mathematical Monographs 23. Amer. Math. Soc., Providence, RI. Translated from the Russian by S. Smith.
  • [24] Lorenz, J., Battiston, S. and Schweitzer, F. (2009). Systemic risk in a unifying framework for cascading processes on networks. Eur. Phys. J. B 71 441–460.
  • [25] May, R. M. and Arinaminpathy, N. (2010). Systemic risk: The dynamics of model banking systems. J. R. Soc. Interface 7 823–838.
  • [26] Shin, H. S., Hahm, J.-H. and Shin, K. (2011). Non-core bank liabilities and financial vulnerability. Technical report, Princeton Univ., Princeton, NJ.
  • [27] Skorohod, A. V. (1956). Limit theorems for stochastic processes. Teor. Veroyatn. Primen. 1 289–319.
  • [28] Sly, A. (2016). On one-dimensional multi-particle diffusion limited aggregation. ArXiv:1609.08107.
  • [29] Sznitman, A.-S. (1991). Topics in propagation of chaos. In École D’Été de Probabilités de Saint-Flour XIX—1989. Lecture Notes in Math. 1464 165–251. Springer, Berlin.
  • [30] Watts, D. J. (2002). A simple model of global cascades on random networks. Proc. Natl. Acad. Sci. USA 99 5766–5771.
  • [31] Whitt, W. (2002). Stochastic-Process Limits: An Introduction to Stochastic-Process Limits and Their Application to Queues. Springer, New York.