IDEAS home Printed from https://ideas.repec.org/a/spr/metcap/v12y2010i4d10.1007_s11009-009-9158-y.html
   My bibliography  Save this article

Numerical Computation of First-Passage Times of Increasing Lévy Processes

Author

Listed:
  • Mark Veillette

    (Boston University)

  • Murad S. Taqqu

    (Boston University)

Abstract

Let {D(s), s ≥ 0} be a non-decreasing Lévy process. The first-hitting time process {E(t), t ≥ 0} (which is sometimes referred to as an inverse subordinator) defined by $E(t) = \inf \{s: D(s) > t \}$ is a process which has arisen in many applications. Of particular interest is the mean first-hitting time $U(t)=\mathbb{E}E(t)$ . This function characterizes all finite-dimensional distributions of the process E. The function U can be calculated by inverting the Laplace transform of the function $\widetilde{U}(\lambda) = (\lambda \phi(\lambda))^{-1}$ , where ϕ is the Lévy exponent of the subordinator D. In this paper, we give two methods for computing numerically the inverse of this Laplace transform. The first is based on the Bromwich integral and the second is based on the Post-Widder inversion formula. The software written to support this work is available from the authors and we illustrate its use at the end of the paper.

Suggested Citation

  • Mark Veillette & Murad S. Taqqu, 2010. "Numerical Computation of First-Passage Times of Increasing Lévy Processes," Methodology and Computing in Applied Probability, Springer, vol. 12(4), pages 695-729, December.
    • Handle: RePEc:spr:metcap:v:12:y:2010:i:4:d:10.1007_s11009-009-9158-y
    DOI: 10.1007/s11009-009-9158-y
    as

    Download full text from publisher

    File URL: http://link.springer.com/10.1007/s11009-009-9158-y
    File Function: Abstract
    Download Restriction: Access to the full text of the articles in this series is restricted.
    File URL: https://libkey.io/10.1007/s11009-009-9158-y?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. Eberlein, Ernst & Keller, Ulrich & Prause, Karsten, 1998. "New Insights into Smile, Mispricing, and Value at Risk: The Hyperbolic Model," The Journal of Business, University of Chicago Press, vol. 71(3), pages 371-405, July.
    2. Meerschaert, Mark M. & Scheffler, Hans-Peter, 2006. "Stochastic model for ultraslow diffusion," Stochastic Processes and their Applications, Elsevier, vol. 116(9), pages 1215-1235, September.
    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. K. K. Kataria & M. Khandakar, 2021. "On the Long-Range Dependence of Mixed Fractional Poisson Process," Journal of Theoretical Probability, Springer, vol. 34(3), pages 1607-1622, September.
    2. Dexter O. Cahoy & Federico Polito & Vir Phoha, 2015. "Transient Behavior of Fractional Queues and Related Processes," Methodology and Computing in Applied Probability, Springer, vol. 17(3), pages 739-759, September.
    3. A. Maheshwari & P. Vellaisamy, 2019. "Fractional Poisson Process Time-Changed by Lévy Subordinator and Its Inverse," Journal of Theoretical Probability, Springer, vol. 32(3), pages 1278-1305, September.
    4. Dhar Soma & Mahanta Lipi B. & Das Kishore Kumar, 2019. "Formulation Of The Simple Markovian Model Using Fractional Calculus Approach And Its Application To Analysis Of Queue Behaviour Of Severe Patients," Statistics in Transition New Series, Statistics Poland, vol. 20(1), pages 117-129, March.

    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. Peter Carr & Liuren Wu, 2014. "Static Hedging of Standard Options," Journal of Financial Econometrics, Oxford University Press, vol. 12(1), pages 3-46.
    2. Yeap, Claudia & Kwok, Simon S. & Choy, S. T. Boris, 2016. "A Flexible Generalised Hyperbolic Option Pricing Model and its Special Cases," Working Papers 2016-14, University of Sydney, School of Economics.
    3. Constantinos Kardaras, 2009. "No‐Free‐Lunch Equivalences For Exponential Lévy Models Under Convex Constraints On Investment," Mathematical Finance, Wiley Blackwell, vol. 19(2), pages 161-187, April.
    4. Veillette, Mark & Taqqu, Murad S., 2010. "Using differential equations to obtain joint moments of first-passage times of increasing Lévy processes," Statistics & Probability Letters, Elsevier, vol. 80(7-8), pages 697-705, April.
    5. Martijn Pistorius & Johannes Stolte, 2012. "Fast computation of vanilla prices in time-changed models and implied volatilities using rational approximations," Papers 1203.6899, arXiv.org.
    6. Geman, Helyette, 2002. "Pure jump Levy processes for asset price modelling," Journal of Banking & Finance, Elsevier, vol. 26(7), pages 1297-1316, July.
    7. Nakajima, Jouchi & Omori, Yasuhiro, 2012. "Stochastic volatility model with leverage and asymmetrically heavy-tailed error using GH skew Student’s t-distribution," Computational Statistics & Data Analysis, Elsevier, vol. 56(11), pages 3690-3704.
    8. Jos� Fajardo & Ernesto Mordecki, 2014. "Skewness premium with L�vy processes," Quantitative Finance, Taylor & Francis Journals, vol. 14(9), pages 1619-1626, September.
    9. Gehrig, Thomas & Iannino, Maria Chiara, 2021. "Did the Basel Process of capital regulation enhance the resiliency of European banks?," Journal of Financial Stability, Elsevier, vol. 55(C).
    10. Calvet, Laurent E. & Fisher, Adlai J., 2008. "Multifrequency jump-diffusions: An equilibrium approach," Journal of Mathematical Economics, Elsevier, vol. 44(2), pages 207-226, January.
    11. Hussein Khraibani & Bilal Nehme & Olivier Strauss, 2018. "Interval Estimation of Value-at-Risk Based on Nonparametric Models," Econometrics, MDPI, vol. 6(4), pages 1-30, December.
    12. Eling, Martin, 2014. "Fitting asset returns to skewed distributions: Are the skew-normal and skew-student good models?," Insurance: Mathematics and Economics, Elsevier, vol. 59(C), pages 45-56.
    13. Yoshio Miyahara & Alexander Novikov, 2001. "Geometric Lévy Process Pricing Model," Research Paper Series 66, Quantitative Finance Research Centre, University of Technology, Sydney.
    14. Arismendi, Juan C. & Broda, Simon, 2017. "Multivariate elliptical truncated moments," Journal of Multivariate Analysis, Elsevier, vol. 157(C), pages 29-44.
    15. Laura Ballota & Griselda Deelstra & Grégory Rayée, 2015. "Quanto Implied Correlation in a Multi-Lévy Framework," Working Papers ECARES ECARES 2015-36, ULB -- Universite Libre de Bruxelles.
    16. Lele Yuan & Kewei Liang & Huidi Wang, 2023. "Solving Inverse Problem of Distributed-Order Time-Fractional Diffusion Equations Using Boundary Observations and L 2 Regularization," Mathematics, MDPI, vol. 11(14), pages 1-20, July.
    17. Wong, Woon K., 2008. "Backtesting trading risk of commercial banks using expected shortfall," Journal of Banking & Finance, Elsevier, vol. 32(7), pages 1404-1415, July.
    18. Weron, Rafał, 2004. "Computationally intensive Value at Risk calculations," Papers 2004,32, Humboldt University of Berlin, Center for Applied Statistics and Economics (CASE).
    19. Li, Minqiang, 2008. "Price Deviations of S&P 500 Index Options from the Black-Scholes Formula Follow a Simple Pattern," MPRA Paper 11530, University Library of Munich, Germany.
    20. Eling, Martin, 2012. "Fitting insurance claims to skewed distributions: Are the skew-normal and skew-student good models?," Insurance: Mathematics and Economics, Elsevier, vol. 51(2), pages 239-248.

    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:spr:metcap:v:12:y:2010:i:4:d:10.1007_s11009-009-9158-y. 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: Sonal Shukla or Springer Nature Abstracting and Indexing (email available below). General contact details of provider: http://www.springer.com .

    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.