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A multilevel Monte Carlo algorithm for Lévy-driven stochastic differential equations

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  • Dereich, Steffen
  • Heidenreich, Felix

Abstract

This article introduces and analyzes multilevel Monte Carlo schemes for the evaluation of the expectation , where Y=(Yt)t[set membership, variant][0,1] is a solution of a stochastic differential equation driven by a Lévy process. Upper bounds are provided for the worst case error over the class of all path dependent measurable functions f, which are Lipschitz continuous with respect to the supremum norm. In the case where the Blumenthal-Getoor index of the driving process is smaller than one, one obtains convergence rates of order , when the computational cost n tends to infinity. This rate is optimal up to logarithms in the case where Y is itself a Lévy process. Furthermore, an error estimate for Blumenthal-Getoor indices larger than one is included together with results of numerical experiments.

Suggested Citation

  • 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.
    • Handle: RePEc:eee:spapps:v:121:y:2011:i:7:p:1565-1587
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    References listed on IDEAS

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    1. Rubenthaler, Sylvain, 2003. "Numerical simulation of the solution of a stochastic differential equation driven by a Lévy process," Stochastic Processes and their Applications, Elsevier, vol. 103(2), pages 311-349, February.
    2. Rainer Avikainen, 2009. "On irregular functionals of SDEs and the Euler scheme," Finance and Stochastics, Springer, vol. 13(3), pages 381-401, September.
    3. Michael B. Giles, 2008. "Multilevel Monte Carlo Path Simulation," Operations Research, INFORMS, vol. 56(3), pages 607-617, June.
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    Citations

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    Cited by:

    1. Michael B. Giles & Yuan Xia, 2017. "Multilevel Monte Carlo for exponential Lévy models," Finance and Stochastics, Springer, vol. 21(4), pages 995-1026, October.
    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. Michael B. Giles & Kristian Debrabant & Andreas Ro{ss}ler, 2013. "Analysis of multilevel Monte Carlo path simulation using the Milstein discretisation," Papers 1302.4676, arXiv.org, revised Jun 2019.
    4. Przybyłowicz, Paweł & Szölgyenyi, Michaela, 2021. "Existence, uniqueness, and approximation of solutions of jump-diffusion SDEs with discontinuous drift," Applied Mathematics and Computation, Elsevier, vol. 403(C).
    5. Mike Giles & Lukasz Szpruch, 2012. "Multilevel Monte Carlo methods for applications in finance," Papers 1212.1377, arXiv.org.
    6. 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.
    7. 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.
    8. 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.
    9. Desmond J. Higham, 2015. "An Introduction to Multilevel Monte Carlo for Option Valuation," Papers 1505.00965, arXiv.org.
    10. 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.
    11. Ahmed Kebaier & Jérôme Lelong, 2018. "Coupling Importance Sampling and Multilevel Monte Carlo using Sample Average Approximation," Post-Print hal-01214840, HAL.
    12. Mike Giles & Yuan Xia, 2014. "Multilevel Monte Carlo For Exponential L\'{e}vy Models," Papers 1403.5309, arXiv.org, revised May 2017.
    13. Duan, Wei-Long & Fang, Hui & Zeng, Chunhua, 2019. "Second-order algorithm for simulating stochastic differential equations with white noises," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 525(C), pages 491-497.
    14. Yuan Xia, 2011. "Multilevel Monte Carlo method for jump-diffusion SDEs," Papers 1106.4730, arXiv.org.
    15. 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.

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