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Propagation of minimality in the supercooled Stefan problem

Author

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  • Christa Cuchiero
  • Stefan Rigger
  • Sara Svaluto-Ferro

Abstract

Supercooled Stefan problems describe the evolution of the boundary between the solid and liquid phases of a substance, where the liquid is assumed to be cooled below its freezing point. Following the methodology of Delarue, Nadtochiy and Shkolnikov, we construct solutions to the one-phase one-dimensional supercooled Stefan problem through a certain McKean-Vlasov equation, which allows to define global solutions even in the presence of blow-ups. Solutions to the McKean-Vlasov equation arise as mean-field limits of particle systems interacting through hitting times, which is important for systemic risk modeling. Our main contributions are: (i) we prove a general tightness theorem for the Skorokhod M1-topology which applies to processes that can be decomposed into a continuous and a monotone part. (ii) We prove propagation of chaos for a perturbed version of the particle system for general initial conditions. (iii) We prove a conjecture of Delarue, Nadtochiy and Shkolnikov, relating the solution concepts of so-called minimal and physical solutions, showing that minimal solutions of the McKean-Vlasov equation are physical whenever the initial condition is integrable.

Suggested Citation

  • Christa Cuchiero & Stefan Rigger & Sara Svaluto-Ferro, 2020. "Propagation of minimality in the supercooled Stefan problem," Papers 2010.03580, arXiv.org, revised Jun 2022.
    • Handle: RePEc:arx:papers:2010.03580
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    References listed on IDEAS

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    1. Ben Hambly & Andreas Søjmark, 2019. "An SPDE model for systemic risk with endogenous contagion," Finance and Stochastics, Springer, vol. 23(3), pages 535-594, July.
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    Cited by:

    1. Zachary Feinstein & Andreas Sojmark, 2021. "Contagious McKean-Vlasov systems with heterogeneous impact and exposure," Papers 2104.06776, arXiv.org, revised Sep 2022.
    2. Erhan Bayraktar & Gaoyue Guo & Wenpin Tang & Yuming Zhang, 2020. "McKean-Vlasov equations involving hitting times: blow-ups and global solvability," Papers 2010.14646, arXiv.org, revised Jul 2023.

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