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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Abstract

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

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

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

Permanent link to this document
https://projecteuclid.org/euclid.aoap/1544000426

Digital Object Identifier
doi:10.1214/18-AAP1403

Mathematical Reviews number (MathSciNet)
MR3910001

Zentralblatt MATH identifier
07039122

Subjects
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)

Keywords
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

Citation

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. https://projecteuclid.org/euclid.aoap/1544000426


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