IDEAS home Printed from https://ideas.repec.org/a/eee/spapps/v132y2021icp76-107.html
   My bibliography  Save this article

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

Author

Listed:
  • 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.

Suggested Citation

  • 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
    as

    Download full text from publisher

    File URL: http://www.sciencedirect.com/science/article/pii/S0304414920303926
    Download Restriction: Full text for ScienceDirect subscribers only
    File URL: https://libkey.io/10.1016/j.spa.2020.10.002?utm_source=ideas
    LibKey link: if access is restricted and if your library uses this service, LibKey will redirect you to where you can use your library subscription to access this item
    ---><---

    As the access to this document is restricted, you may want to search for a different version of it.

    References listed on IDEAS

    as
    1. 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.
    2. Ying Jiao & Chunhua Ma & Simone Scotti, 2017. "Alpha-CIR Model with Branching Processes in Sovereign Interest Rate Modelling," Post-Print hal-01275397, HAL.
    3. 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.
    4. 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.
    5. 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.
    6. Claudio Fontana & Alessandro Gnoatto & Guillaume Szulda, 2019. "Multiple Yield Curve Modelling with CBI Processes," Working Papers 19/2019, University of Verona, Department of Economics.
    7. 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.
    8. 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.
    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.
    Full references (including those not matched with items on IDEAS)

    Most related items

    These are the items that most often cite the same works as this one and are cited by the same works as this one.
    1. Fontana, Claudio & Gnoatto, Alessandro & Szulda, Guillaume, 2023. "CBI-time-changed Lévy processes," Stochastic Processes and their Applications, Elsevier, vol. 163(C), pages 323-349.
    2. Claudio Fontana & Alessandro Gnoatto & Guillaume Szulda, 2022. "CBI-time-changed Lévy processes," Working Papers 05/2022, University of Verona, Department of Economics.
    3. Claudio Fontana & Alessandro Gnoatto & Guillaume Szulda, 2021. "CBI-time-changed Lévy processes for multi-currency modeling," Working Papers 14/2021, University of Verona, Department of Economics.
    4. Agostino Capponi & Xu Sun & David D. Yao, 2020. "A Dynamic Network Model of Interbank Lending—Systemic Risk and Liquidity Provisioning," Mathematics of Operations Research, INFORMS, vol. 45(3), pages 1127-1152, August.
    5. Jean-David Fermanian, 2020. "On the Dependence between Default Risk and Recovery Rates in Structural Models," Annals of Economics and Statistics, GENES, issue 140, pages 45-82.
    6. Egami, M. & Kevkhishvili, R., 2017. "An analysis of simultaneous company defaults using a shot noise process," Journal of Banking & Finance, Elsevier, vol. 80(C), pages 135-161.
    7. Micha{l} Barski & Rafa{l} {L}ochowski, 2024. "Affine term structure models driven by independent L\'evy processes," Papers 2402.07503, arXiv.org.
    8. Matyas Barczy & Mohamed Ben Alaya & Ahmed Kebaier & Gyula Pap, 2017. "Asymptotic properties of maximum likelihood estimator for the growth rate of a stable CIR process based on continuous time observations," Papers 1711.02140, arXiv.org, revised Feb 2019.
    9. Konstantinos Spiliopoulos & Jia Yang, 2018. "Network effects in default clustering for large systems," Papers 1812.07645, arXiv.org, revised Feb 2020.
    10. Burzoni, Matteo & Campi, Luciano, 2023. "Mean field games with absorption and common noise with a model of bank run," Stochastic Processes and their Applications, Elsevier, vol. 164(C), pages 206-241.
    11. Ben Hambly & Nikolaos Kolliopoulos, 2020. "Fast mean-reversion asymptotics for large portfolios of stochastic volatility models," Finance and Stochastics, Springer, vol. 24(3), pages 757-794, July.
    12. Ben Hambly & Andreas Sojmark, 2018. "An SPDE Model for Systemic Risk with Endogenous Contagion," Papers 1801.10088, arXiv.org, revised Sep 2018.
    13. Tang, Qihe & Tang, Zhaofeng & Yang, Yang, 2019. "Sharp asymptotics for large portfolio losses under extreme risks," European Journal of Operational Research, Elsevier, vol. 276(2), pages 710-722.
    14. Josselin Garnier & George Papanicolaou & Tzu-Wei Yang, 2015. "A risk analysis for a system stabilized by a central agent," Papers 1507.08333, arXiv.org, revised Aug 2015.
    15. Sirignano, Justin & Spiliopoulos, Konstantinos, 2020. "Mean field analysis of neural networks: A central limit theorem," Stochastic Processes and their Applications, Elsevier, vol. 130(3), pages 1820-1852.
    16. Justin Sirignano & Kay Giesecke, 2019. "Risk Analysis for Large Pools of Loans," Management Science, INFORMS, vol. 65(1), pages 107-121, January.
    17. 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.
    18. Pingping Jiang & Bo Li & Yongjin Wang, 2020. "Exit Times, Undershoots and Overshoots for Reflected CIR Process with Two-Sided Jumps," Methodology and Computing in Applied Probability, Springer, vol. 22(2), pages 693-710, June.
    19. Li, Libo & Taguchi, Dai, 2019. "On the Euler–Maruyama scheme for spectrally one-sided Lévy driven SDEs with Hölder continuous coefficients," Statistics & Probability Letters, Elsevier, vol. 146(C), pages 15-26.
    20. Tang, Qihe & Tong, Zhiwei & Yang, Yang, 2021. "Large portfolio losses in a turbulent market," European Journal of Operational Research, Elsevier, vol. 292(2), pages 755-769.

    Corrections

    All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:eee:spapps:v:132:y:2021:i:c:p:76-107. See general information about how to correct material in RePEc.

    If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

    If CitEc recognized a bibliographic reference but did not link an item in RePEc to it, you can help with this form .

    If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: Catherine Liu (email available below). General contact details of provider: http://www.elsevier.com/wps/find/journaldescription.cws_home/505572/description#description .

    Please note that corrections may take a couple of weeks to filter through the various RePEc services.

    IDEAS is a RePEc service. RePEc uses bibliographic data supplied by the respective publishers.