TY - JOUR

T1 - Mean field systems on networks, with singular interaction through hitting times

AU - Nadtochiy, Sergey

AU - Shkolnikov, Mykhaylo

N1 - Funding Information:
Consequently, with {kx}x∈X obeying minx∈X(kx/g′(P(τx > t))) > e−ρ, the summands in the series of (B.2) grow exponentially and the result readily follows from Lemma B.1. (b) The assumptions of Theorem 3.3(b) imply the existence of an eigenvector v = (vx)x∈X of the matrix A = (C(x)κ(x,{x′})cx′)x,x′∈X with strictly positive entries such that Av < v (see, e.g., [22], Theorem 3.1.1(c)). Then ηnx := vx/n satisfy (2.10) for all large enough n. Hence, Theorem 2.3 and Remark 2.4 imply the existence of a solution to (2.1), with the prescribed distribution at t− and continuous at t. □ Acknowledgments. The first author was supported in part by the NSF CAREER Grant DMS-1855309. The second author was supported in part by the NSF Grant DMS-1506290 and a Princeton SEAS innovation research grant.

PY - 2020

Y1 - 2020

N2 - Building on the line of work (Ann. Appl. Probab. 25 (2015) 2096-2133; Stochastic Process. Appl. 125 (2015) 2451-2492; Ann. Appl. Probab. 29 (2019) 89-129; Arch. Ration. Mech. Anal. 233 (2019) 643-699; Ann. Appl. Probab. 29 (2019) 2338-2373; Finance Stoch. 23 (2019) 535-594), we continue the study of particle systems with singular interaction through hitting times. In contrast to the previous research, we (i) consider very general driving processes and interaction functions, (ii) allow for inhomogeneous connection structures and (iii) analyze a game in which the particles determine their connections strategically. Hereby, we uncover two completely new phenomena. First, we characterize the "times of fragility" of such systems (e.g., the times when a macroscopic part of the population defaults or gets infected simultaneously, or when the neuron cells "synchronize") explicitly in terms of the dynamics of the driving processes, the current distribution of the particles' values and the topology of the underlying network (represented by its Perron-Frobenius eigenvalue). Second, we use such systems to describe a dynamic credit-network game and show that, in equilibrium, the system regularizes, that is, the times of fragility never occur, as the particles avoid them by adjusting their connections strategically. Two auxiliary mathematical results, useful in their own right, are uncovered during our investigation: a generalization of Schauder's fixed-point theorem for the Skorokhod space with the M1 topology, and the application of max-plus algebra to the equilibrium version of the network flow problem.

AB - Building on the line of work (Ann. Appl. Probab. 25 (2015) 2096-2133; Stochastic Process. Appl. 125 (2015) 2451-2492; Ann. Appl. Probab. 29 (2019) 89-129; Arch. Ration. Mech. Anal. 233 (2019) 643-699; Ann. Appl. Probab. 29 (2019) 2338-2373; Finance Stoch. 23 (2019) 535-594), we continue the study of particle systems with singular interaction through hitting times. In contrast to the previous research, we (i) consider very general driving processes and interaction functions, (ii) allow for inhomogeneous connection structures and (iii) analyze a game in which the particles determine their connections strategically. Hereby, we uncover two completely new phenomena. First, we characterize the "times of fragility" of such systems (e.g., the times when a macroscopic part of the population defaults or gets infected simultaneously, or when the neuron cells "synchronize") explicitly in terms of the dynamics of the driving processes, the current distribution of the particles' values and the topology of the underlying network (represented by its Perron-Frobenius eigenvalue). Second, we use such systems to describe a dynamic credit-network game and show that, in equilibrium, the system regularizes, that is, the times of fragility never occur, as the particles avoid them by adjusting their connections strategically. Two auxiliary mathematical results, useful in their own right, are uncovered during our investigation: a generalization of Schauder's fixed-point theorem for the Skorokhod space with the M1 topology, and the application of max-plus algebra to the equilibrium version of the network flow problem.

KW - Cascades

KW - Credit network game

KW - Directed weighted graphs

KW - Dynamic games

KW - M1 topology

KW - Max-plus algebra

KW - Mean field games on graphs

KW - Nash equilibrium

KW - Network flow problem

KW - Particle systems

KW - Perron-frobenius eigenvalue

KW - Regularization through a game

KW - Schauder's fixed-point theorem

KW - Self-excitation

KW - Singular interaction through hitting times

KW - Systemic risk

KW - Times of fragility

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UR - http://www.scopus.com/inward/citedby.url?scp=85089245154&partnerID=8YFLogxK

U2 - 10.1214/19-AOP1403

DO - 10.1214/19-AOP1403

M3 - Article

AN - SCOPUS:85089245154

VL - 48

SP - 1520

EP - 1556

JO - Annals of Probability

JF - Annals of Probability

SN - 0091-1798

IS - 3

ER -