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An SPDE with Robin-type boundary for a system of elastically killed diffusions on the positive half-line

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  • Hambly, Ben
  • Meier, Julian
  • Søjmark, Andreas

Abstract

We consider a system of particles undergoing correlated diffusion with elastic boundary conditions on the half-line in the limit as the number of particles goes to infinity. We establish existence and uniqueness for the limiting empirical measure valued process for the surviving particles, which is a weak form for an SPDE with a noisy Robin boundary condition satisfied by the particle density. We show that this density process has good L2-regularity properties in the interior of the domain but may exhibit singularities on the boundary at a dense set of times. We make connections to the corresponding absorbing and reflecting SPDEs as the elastic parameter varies.

Suggested Citation

  • Hambly, Ben & Meier, Julian & Søjmark, Andreas, 2025. "An SPDE with Robin-type boundary for a system of elastically killed diffusions on the positive half-line," Stochastic Processes and their Applications, Elsevier, vol. 179(C).
    • Handle: RePEc:eee:spapps:v:179:y:2025:i:c:s030441492400228x
    DOI: 10.1016/j.spa.2024.104520
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    References listed on IDEAS

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    1. Kurtz, Thomas G. & Xiong, Jie, 1999. "Particle representations for a class of nonlinear SPDEs," Stochastic Processes and their Applications, Elsevier, vol. 83(1), pages 103-126, September.
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    3. Ahmad, F. & Hambly, B.M. & Ledger, S., 2018. "A stochastic partial differential equation model for the pricing of mortgage-backed securities," Stochastic Processes and their Applications, Elsevier, vol. 128(11), pages 3778-3806.
    4. 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.
    5. Kay Giesecke & Konstantinos Spiliopoulos & Richard B. Sowers, 2011. "Default clustering in large portfolios: Typical events," Papers 1104.1773, arXiv.org, revised Feb 2013.
    6. Michael B. Giles & Christoph Reisinger, 2012. "Stochastic finite differences and multilevel Monte Carlo for a class of SPDEs in finance," Papers 1204.1442, arXiv.org.
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