TY - JOUR

T1 - Particle systems with singular interaction through hitting times

T2 - Application in systemic risk modeling

AU - Nadtochiy, Sergey

AU - Shkolnikov, Mykhaylo

N1 - Funding Information:
Received May 2017; revised November 2017. 1Supported in part by NSF Grant DMS-1411824. 2Supported in part by NSF Grant DMS-1506290. MSC2010 subject classifications. 35B65, 35K20, 82C22, 91G80. Key words and phrases. 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.
Publisher Copyright:
© Institute of Mathematical Statistics, 2019.

PY - 2019/2

Y1 - 2019/2

N2 - 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.

AB - 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.

KW - Banking systems

KW - Blow-ups in parabolic partial differential equations

KW - Default cascades

KW - Interacting particle systems

KW - Large system limits

KW - Loss of continuity

KW - Mean-field models

KW - Noncore exposures

KW - Nonlinear cauchy–dirichlet problems

KW - Regularity estimates

KW - Self-excitation

KW - Singular interaction

KW - Systemic crises

KW - Systemic risk

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U2 - 10.1214/18-AAP1403

DO - 10.1214/18-AAP1403

M3 - Article

AN - SCOPUS:85058455647

SN - 1050-5164

VL - 29

SP - 89

EP - 129

JO - Annals of Applied Probability

JF - Annals of Applied Probability

IS - 1

ER -