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Well-posedness and approximation of some one-dimensional Lévy-driven non-linear SDEs

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  • Frikha, Noufel
  • Li, Libo

Abstract

In this article, we are interested in the strong well-posedness together with the numerical approximation of some one-dimensional stochastic differential equations with a non-linear drift, in the sense of McKean–Vlasov, driven by a spectrally-positive Lévy process and a Brownian motion. We provide criteria for the existence of strong solutions under non-Lipschitz conditions of Yamada–Watanabe type without non-degeneracy assumption following the approach developed by Li and Mytnik (2011). The strong convergence rate of the propagation of chaos for the associated particle system and of the corresponding Euler–Maruyama scheme are also investigated. In particular, the strong convergence rate of the Euler–Maruyama scheme exhibits an interplay between the regularity of the coefficients and the order of singularity of the Lévy measure around zero.

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  • Frikha, Noufel & Li, Libo, 2021. "Well-posedness and approximation of some one-dimensional Lévy-driven non-linear SDEs," Stochastic Processes and their Applications, Elsevier, vol. 132(C), pages 76-107.
    • Handle: RePEc:eee:spapps:v:132:y:2021:i:c:p:76-107
    DOI: 10.1016/j.spa.2020.10.002
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    References listed on IDEAS

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    1. Kay Giesecke & Konstantinos Spiliopoulos & Richard B. Sowers & Justin A. Sirignano, 2015. "Large Portfolio Asymptotics For Loss From Default," Mathematical Finance, Wiley Blackwell, vol. 25(1), pages 77-114, January.
    2. Jiao, Ying & Ma, Chunhua & Scotti, Simone & Sgarra, Carlo, 2019. "A branching process approach to power markets," Energy Economics, Elsevier, vol. 79(C), pages 144-156.
    3. Claudio Fontana & Alessandro Gnoatto & Guillaume Szulda, 2019. "Multiple Yield Curve Modelling with CBI Processes," Working Papers 19/2019, University of Verona, Department of Economics.
    4. Chaudru de Raynal, P.E., 2020. "Strong well posedness of McKean–Vlasov stochastic differential equations with Hölder drift," Stochastic Processes and their Applications, Elsevier, vol. 130(1), pages 79-107.
    5. Kay Giesecke & Konstantinos Spiliopoulos & Richard B. Sowers & Justin A. Sirignano, 2011. "Large Portfolio Asymptotics for Loss From Default," Papers 1109.1272, arXiv.org, revised Feb 2015.
    6. Ying Jiao & Chunhua Ma & Simone Scotti, 2017. "Alpha-CIR model with branching processes in sovereign interest rate modeling," Finance and Stochastics, Springer, vol. 21(3), pages 789-813, July.
    7. Ying Jiao & Chunhua Ma & Simone Scotti, 2017. "Alpha-CIR Model with Branching Processes in Sovereign Interest Rate Modelling," Post-Print hal-01275397, HAL.
    8. Fu, Zongfei & Li, Zenghu, 2010. "Stochastic equations of non-negative processes with jumps," Stochastic Processes and their Applications, Elsevier, vol. 120(3), pages 306-330, March.
    9. Kay Giesecke & Konstantinos Spiliopoulos & Richard B. Sowers, 2011. "Default clustering in large portfolios: Typical events," Papers 1104.1773, arXiv.org, revised Feb 2013.
    10. Ying Jiao & Chunhua Ma & Simone Scotti & Carlo Sgarra, 2019. "A branching process approach to power markets," Post-Print hal-02954986, HAL.
    11. Пигнастый, Олег & Koжевников, Георгий, 2019. "Распределенная Динамическая Pde-Модель Программного Управления Загрузкой Технологического Оборудования Производственной Линии [Distributed dynamic PDE-model of a program control by utilization of t," MPRA Paper 93278, University Library of Munich, Germany, revised 02 Feb 2019.
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    Cited by:

    1. Jie, Lijuan & Luo, Liangqing & Zhang, Hua, 2024. "One-dimensional McKean–Vlasov stochastic Volterra equations with Hölder diffusion coefficients," Statistics & Probability Letters, Elsevier, vol. 205(C).

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