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

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  • Sergey Nadtochiy
  • Mykhaylo Shkolnikov

Abstract

Building on the line of work [DIRT15a], [DIRT15b], [NS17a], [DT17], [HLS18], [HS18] 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: i.e., 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 the max-plus algebra to the equilibrium version of the network flow problem.

Suggested Citation

  • Sergey Nadtochiy & Mykhaylo Shkolnikov, 2018. "Mean field systems on networks, with singular interaction through hitting times," Papers 1807.02015, arXiv.org, revised Sep 2019.
  • Handle: RePEc:arx:papers:1807.02015
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    File URL: http://arxiv.org/pdf/1807.02015
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    References listed on IDEAS

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    1. Andrew G. Atkeson & Andrea L. Eisfeldt & Pierre‐Olivier Weill, 2015. "Entry and Exit in OTC Derivatives Markets," Econometrica, Econometric Society, vol. 83, pages 2231-2292, November.
    2. Ben Hambly & Andreas Sojmark, 2018. "An SPDE Model for Systemic Risk with Endogenous Contagion," Papers 1801.10088, arXiv.org, revised Sep 2018.
    3. Budhiraja, Amarjit & Wu, Ruoyu, 2016. "Some fluctuation results for weakly interacting multi-type particle systems," Stochastic Processes and their Applications, Elsevier, vol. 126(8), pages 2253-2296.
    4. Babus, Ana & Hu, Tai-Wei, 2017. "Endogenous intermediation in over-the-counter markets," Journal of Financial Economics, Elsevier, vol. 125(1), pages 200-215.
    5. Sergey Nadtochiy & Mykhaylo Shkolnikov, 2017. "Particle systems with singular interaction through hitting times: application in systemic risk modeling," Papers 1705.00691, arXiv.org.
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    Cited by:

    1. Sean Ledger & Andreas Sojmark, 2018. "At the Mercy of the Common Noise: Blow-ups in a Conditional McKean--Vlasov Problem," Papers 1807.05126, arXiv.org, revised Mar 2024.
    2. Feinstein, Zachary & Sojmark, Andreas, 2023. "Contagious McKean–Vlasov systems with heterogeneous impact and exposure," LSE Research Online Documents on Economics 119457, London School of Economics and Political Science, LSE Library.
    3. Alexander Lipton & Vadim Kaushansky & Christoph Reisinger, 2018. "Semi-analytical solution of a McKean-Vlasov equation with feedback through hitting a boundary," Papers 1808.05311, arXiv.org, revised Aug 2018.
    4. Sean Ledger & Andreas Sojmark, 2018. "Uniqueness for contagious McKean--Vlasov systems in the weak feedback regime," Papers 1811.12356, arXiv.org, revised Oct 2019.

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