IDEAS home Printed from https://ideas.repec.org/a/eee/spapps/v142y2021icp407-431.html
   My bibliography  Save this article

Implementable coupling of Lévy process and Brownian motion

Author

Listed:
  • Fomichov, Vladimir
  • González Cázares, Jorge
  • Ivanovs, Jevgenijs

Abstract

We provide a simple algorithm for construction of Brownian paths approximating those of a Lévy process on a finite time interval. It requires knowledge of the Lévy process trajectory on a chosen regular grid and the law of its endpoint, or the ability to simulate from that. This algorithm is based on reordering of Brownian increments, and it can be applied in a recursive manner. We establish an upper bound on the mean squared maximal distance between the paths and determine a suitable mesh size in various asymptotic regimes. The analysis proceeds by reduction to the comonotonic coupling of increments. Applications to model risk and multilevel Monte Carlo are discussed in detail, and numerical examples are provided.

Suggested Citation

  • Fomichov, Vladimir & González Cázares, Jorge & Ivanovs, Jevgenijs, 2021. "Implementable coupling of Lévy process and Brownian motion," Stochastic Processes and their Applications, Elsevier, vol. 142(C), pages 407-431.
    • Handle: RePEc:eee:spapps:v:142:y:2021:i:c:p:407-431
    DOI: 10.1016/j.spa.2021.09.008
    as

    Download full text from publisher

    File URL: http://www.sciencedirect.com/science/article/pii/S0304414921001502
    Download Restriction: Full text for ScienceDirect subscribers only
    File URL: https://libkey.io/10.1016/j.spa.2021.09.008?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. Ferreiro-Castilla, A. & Kyprianou, A.E. & Scheichl, R. & Suryanarayana, G., 2014. "Multilevel Monte Carlo simulation for Lévy processes based on the Wiener–Hopf factorisation," Stochastic Processes and their Applications, Elsevier, vol. 124(2), pages 985-1010.
    2. Michael B. Giles & Yuan Xia, 2017. "Multilevel Monte Carlo for exponential Lévy models," Finance and Stochastics, Springer, vol. 21(4), pages 995-1026, October.
    3. Breuer, Thomas & Csiszár, Imre, 2013. "Systematic stress tests with entropic plausibility constraints," Journal of Banking & Finance, Elsevier, vol. 37(5), pages 1552-1559.
    4. Majka, Mateusz B., 2017. "Coupling and exponential ergodicity for stochastic differential equations driven by Lévy processes," Stochastic Processes and their Applications, Elsevier, vol. 127(12), pages 4083-4125.
    5. Mike Giles & Yuan Xia, 2014. "Multilevel Monte Carlo For Exponential L\'{e}vy Models," Papers 1403.5309, arXiv.org, revised May 2017.
    6. Jorge Ignacio Gonz'alez C'azares & Aleksandar Mijatovi'c, 2021. "Monte Carlo algorithm for the extrema of tempered stable processes," Papers 2103.15310, arXiv.org, revised Dec 2022.
    7. Paul Embrechts & Marco Frei, 2009. "Panjer recursion versus FFT for compound distributions," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 69(3), pages 497-508, July.
    8. Jose Blanchet & Karthyek Murthy, 2019. "Quantifying Distributional Model Risk via Optimal Transport," Mathematics of Operations Research, INFORMS, vol. 44(2), pages 565-600, May.
    9. Dereich, Steffen & Heidenreich, Felix, 2011. "A multilevel Monte Carlo algorithm for Lévy-driven stochastic differential equations," Stochastic Processes and their Applications, Elsevier, vol. 121(7), pages 1565-1587, July.
    10. Böttcher, Björn & Schilling, René L. & Wang, Jian, 2011. "Constructions of coupling processes for Lévy processes," Stochastic Processes and their Applications, Elsevier, vol. 121(6), pages 1201-1216, June.
    Full references (including those not matched with items on IDEAS)

    Citations

    Citations are extracted by the CitEc Project, subscribe to its RSS feed for this item.
    as


    Cited by:

    1. Søren Asmussen, 2022. "On the role of skewness and kurtosis in tempered stable (CGMY) Lévy models in finance," Finance and Stochastics, Springer, vol. 26(3), pages 383-416, July.

    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. Jorge Ignacio Gonz'alez C'azares & Aleksandar Mijatovi'c & Ger'onimo Uribe Bravo, 2018. "Geometrically Convergent Simulation of the Extrema of L\'{e}vy Processes," Papers 1810.11039, arXiv.org, revised Jun 2021.
    2. Jorge González Cázares & Aleksandar Mijatović, 2022. "Simulation of the drawdown and its duration in Lévy models via stick-breaking Gaussian approximation," Finance and Stochastics, Springer, vol. 26(4), pages 671-732, October.
    3. Jorge Gonz'alez C'azares & Aleksandar Mijatovi'c, 2020. "Simulation of the drawdown and its duration in L\'{e}vy models via stick-breaking Gaussian approximation," Papers 2011.06618, arXiv.org, revised Mar 2021.
    4. Kathrin Glau & Daniel Kressner & Francesco Statti, 2019. "Low-rank tensor approximation for Chebyshev interpolation in parametric option pricing," Papers 1902.04367, arXiv.org.
    5. Luo, Dejun & Wang, Jian, 2019. "Refined basic couplings and Wasserstein-type distances for SDEs with Lévy noises," Stochastic Processes and their Applications, Elsevier, vol. 129(9), pages 3129-3173.
    6. Kahalé, Nabil, 2020. "General multilevel Monte Carlo methods for pricing discretely monitored Asian options," European Journal of Operational Research, Elsevier, vol. 287(2), pages 739-748.
    7. Michael B. Giles & Yuan Xia, 2017. "Multilevel Monte Carlo for exponential Lévy models," Finance and Stochastics, Springer, vol. 21(4), pages 995-1026, October.
    8. Søren Asmussen & Jevgenijs Ivanovs, 2018. "Discretization error for a two-sided reflected Lévy process," Queueing Systems: Theory and Applications, Springer, vol. 89(1), pages 199-212, June.
    9. Jorge Ignacio Gonz'alez C'azares & Aleksandar Mijatovi'c, 2021. "Monte Carlo algorithm for the extrema of tempered stable processes," Papers 2103.15310, arXiv.org, revised Dec 2022.
    10. Nabil Kahale, 2018. "General multilevel Monte Carlo methods for pricing discretely monitored Asian options," Papers 1805.09427, arXiv.org, revised Sep 2018.
    11. Jianhai Bao & Jian Wang, 2023. "Coupling methods and exponential ergodicity for two‐factor affine processes," Mathematische Nachrichten, Wiley Blackwell, vol. 296(5), pages 1716-1736, May.
    12. Jie Chen & Liaoyuan Fan & Lingfei Li & Gongqiu Zhang, 2022. "A multidimensional Hilbert transform approach for barrier option pricing and survival probability calculation," Review of Derivatives Research, Springer, vol. 25(2), pages 189-232, July.
    13. Makam, Vaishno Devi & Millossovich, Pietro & Tsanakas, Andreas, 2021. "Sensitivity analysis with χ2-divergences," Insurance: Mathematics and Economics, Elsevier, vol. 100(C), pages 372-383.
    14. Meng Qi & Ying Cao & Zuo-Jun (Max) Shen, 2022. "Distributionally Robust Conditional Quantile Prediction with Fixed Design," Management Science, INFORMS, vol. 68(3), pages 1639-1658, March.
    15. Blier-Wong, Christopher & Cossette, Hélène & Marceau, Etienne, 2023. "Risk aggregation with FGM copulas," Insurance: Mathematics and Economics, Elsevier, vol. 111(C), pages 102-120.
    16. Peter Grundke & Kamil Pliszka, 2018. "A macroeconomic reverse stress test," Review of Quantitative Finance and Accounting, Springer, vol. 50(4), pages 1093-1130, May.
    17. Anthony Coache & Sebastian Jaimungal, 2024. "Robust Reinforcement Learning with Dynamic Distortion Risk Measures," Papers 2409.10096, arXiv.org.
    18. Fallou Niakh, 2023. "A fixed point approach for computing actuarially fair Pareto optimal risk-sharing rules," Papers 2303.05421, arXiv.org, revised Jul 2023.
    19. Liu, Meng & Bai, Chuanzhi, 2016. "Dynamics of a stochastic one-prey two-predator model with Lévy jumps," Applied Mathematics and Computation, Elsevier, vol. 284(C), pages 308-321.
    20. Pritsker, Matt, 2019. "An overview of regulatory stress-testing and steps to improve it," Global Finance Journal, Elsevier, vol. 39(C), pages 39-43.

    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:142:y:2021:i:c:p:407-431. 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.