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Strong convergence of the Euler–Maruyama approximation for a class of Lévy-driven SDEs

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  • Kühn, Franziska
  • Schilling, René L.

Abstract

Consider the following stochastic differential equation (SDE) dXt=b(t,Xt−)dt+dLt,X0=x,driven by a d-dimensional Lévy process (Lt)t≥0. We establish conditions on the Lévy process and the drift coefficient b such that the Euler–Maruyama approximation converges strongly to a solution of the SDE with an explicitly given rate. The convergence rate depends on the regularity of b and the behaviour of the Lévy measure at the origin. As a by-product of the proof, we obtain that the SDE has a pathwise unique solution. Our result covers many important examples of Lévy processes, e.g. isotropic stable, relativistic stable, tempered stable and layered stable.

Suggested Citation

  • Kühn, Franziska & Schilling, René L., 2019. "Strong convergence of the Euler–Maruyama approximation for a class of Lévy-driven SDEs," Stochastic Processes and their Applications, Elsevier, vol. 129(8), pages 2654-2680.
    • Handle: RePEc:eee:spapps:v:129:y:2019:i:8:p:2654-2680
    DOI: 10.1016/j.spa.2018.07.018
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    References listed on IDEAS

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    1. Deng, Chang-Song & Schilling, René L., 2015. "On shift Harnack inequalities for subordinate semigroups and moment estimates for Lévy processes," Stochastic Processes and their Applications, Elsevier, vol. 125(10), pages 3851-3878.
    2. Küchler, Uwe & Tappe, Stefan, 2013. "Tempered stable distributions and processes," Stochastic Processes and their Applications, Elsevier, vol. 123(12), pages 4256-4293.
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    1. Li, Yanfang & Zhao, Guohuan, 2024. "Euler–Maruyama scheme for SDE driven by Lévy process with Hölder drift," Statistics & Probability Letters, Elsevier, vol. 215(C).
    2. Wu, Mingyan & Hao, Zimo, 2023. "Well-posedness of density dependent SDE driven by α-stable process with Hölder drifts," Stochastic Processes and their Applications, Elsevier, vol. 164(C), pages 416-442.

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