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Uniqueness for contagious McKean--Vlasov systems in the weak feedback regime

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  • Sean Ledger
  • Andreas Sojmark

Abstract

We present a simple uniqueness argument for a collection of McKean-Vlasov problems that have seen recent interest. Our first result shows that, in the weak feedback regime, there is global uniqueness for a very general class of random drivers. By weak feedback we mean the case where the contagion parameters are small enough to prevent blow-ups in solutions. Next, we specialise to a Brownian driver and show how the same techniques can be extended to give short-time uniqueness after blow-ups, regardless of the feedback strength. The heart of our approach is a surprisingly simple probabilistic comparison argument that is robust in the sense that it does not ask for any regularity of the solutions.

Suggested Citation

  • Sean Ledger & Andreas Sojmark, 2018. "Uniqueness for contagious McKean--Vlasov systems in the weak feedback regime," Papers 1811.12356, arXiv.org, revised Oct 2019.
    • Handle: RePEc:arx:papers:1811.12356
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    References listed on IDEAS

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    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. Sergey Nadtochiy & Mykhaylo Shkolnikov, 2018. "Mean field systems on networks, with singular interaction through hitting times," Papers 1807.02015, arXiv.org, revised Sep 2019.
    3. Ben Hambly & Andreas Sojmark, 2018. "An SPDE Model for Systemic Risk with Endogenous Contagion," Papers 1801.10088, arXiv.org, revised Sep 2018.
    4. 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.
    5. 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.
    6. 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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