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At the Mercy of the Common Noise: Blow-ups in a Conditional McKean--Vlasov Problem

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  • Sean Ledger
  • Andreas Sojmark

Abstract

We extend a model of positive feedback and contagion in large mean-field systems, by introducing a common source of noise driven by Brownian motion. Although the driving dynamics are continuous, the positive feedback effect can lead to `blow-up' phenomena whereby solutions develop jump-discontinuities. Our main results are twofold and concern the conditional McKean--Vlasov formulation of the model. First and foremost, we show that there are global solutions to this McKean--Vlasov problem, which can be realised as limit points of a motivating particle system with common noise. Furthermore, we derive results on the occurrence of blow-ups, thereby showing how these events can be triggered or prevented by the pathwise realisations of the common noise.

Suggested Citation

  • 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.
    • Handle: RePEc:arx:papers:1807.05126
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    References listed on IDEAS

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    1. Sergey Nadtochiy & Mykhaylo Shkolnikov, 2018. "Mean field systems on networks, with singular interaction through hitting times," Papers 1807.02015, arXiv.org, revised Sep 2019.
    2. Ben Hambly & Andreas Sojmark, 2018. "An SPDE Model for Systemic Risk with Endogenous Contagion," Papers 1801.10088, arXiv.org, revised Sep 2018.
    3. Delarue, F. & Inglis, J. & Rubenthaler, S. & Tanré, E., 2015. "Particle systems with a singular mean-field self-excitation. Application to neuronal networks," Stochastic Processes and their Applications, Elsevier, vol. 125(6), pages 2451-2492.
    4. 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. 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.
    2. Christa Cuchiero & Christoph Reisinger & Stefan Rigger, 2021. "Optimal bailout strategies resulting from the drift controlled supercooled Stefan problem," Papers 2111.01783, arXiv.org, revised Oct 2022.
    3. Zachary Feinstein & Andreas Søjmark, 2023. "Contagious McKean–Vlasov systems with heterogeneous impact and exposure," Finance and Stochastics, Springer, vol. 27(3), pages 663-711, July.
    4. Zachary Feinstein & Andreas Sojmark, 2021. "Contagious McKean-Vlasov systems with heterogeneous impact and exposure," Papers 2104.06776, arXiv.org, revised Sep 2022.
    5. 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.
    6. 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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