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Statistical inference for time-changed Lévy processes via Mellin transform approach

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  • Belomestny, Denis
  • Schoenmakers, John

Abstract

Given a Lévy process (Lt)t≥0 and an independent nondecreasing process (time change) (T(t))t≥0, we consider the problem of statistical inference on T based on low-frequency observations of the time-changed Lévy process LT(t). Our approach is based on the genuine use of Mellin and Laplace transforms. We propose a consistent estimator for the density of the increments of T in a stationary regime, derive its convergence rates and prove the optimality of the rates. It turns out that the convergence rates heavily depend on the decay of the Mellin transform of T. Finally, the performance of the estimator is analysed via a Monte Carlo simulation study.

Suggested Citation

  • Belomestny, Denis & Schoenmakers, John, 2016. "Statistical inference for time-changed Lévy processes via Mellin transform approach," Stochastic Processes and their Applications, Elsevier, vol. 126(7), pages 2092-2122.
    • Handle: RePEc:eee:spapps:v:126:y:2016:i:7:p:2092-2122
    DOI: 10.1016/j.spa.2016.01.005
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    References listed on IDEAS

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    1. Belomestny, Denis & Schoenmakers, John, 2015. "Statistical Skorohod embedding problem: Optimality and asymptotic normality," Statistics & Probability Letters, Elsevier, vol. 104(C), pages 169-180.
    2. Adam D. Bull, 2013. "Estimating time-changes in noisy L\'evy models," Papers 1312.5911, arXiv.org, revised Nov 2014.
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    Cited by:

    1. P. Carr & A. Itkin & D. Muravey, 2022. "Semi-analytical pricing of barrier options in the time-dependent Heston model," Papers 2202.06177, arXiv.org.
    2. Gery Geenens, 2021. "Mellin–Meijer kernel density estimation on $${{\mathbb {R}}}^+$$ R +," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 73(5), pages 953-977, October.
    3. Shimizu, Yasutaka & Zhang, Zhimin, 2017. "Estimating Gerber–Shiu functions from discretely observed Lévy driven surplus," Insurance: Mathematics and Economics, Elsevier, vol. 74(C), pages 84-98.
    4. Viktor Schulmann, 2021. "Estimation of stopping times for stopped self-similar random processes," Statistical Inference for Stochastic Processes, Springer, vol. 24(2), pages 477-498, July.

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