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Numerical Techniques in Lévy Fluctuation Theory

Author

Listed:
  • Naser M. Asghari

    (University of Amsterdam)

  • Peter Iseger

    (ABN-Amro)

  • Michael Mandjes

    (University of Amsterdam
    Eindhoven University of Technology
    CWI)

Abstract

This paper presents a framework for numerical computations in fluctuation theory for Lévy processes. More specifically, with $\bar X_t:= \sup_{0\le s\le t} X_s$ denoting the running maximum of the Lévy process X t , the aim is to evaluate ${\mathbb P}(\bar X_t \le x)$ for t,x > 0. We do so by approximating the Lévy process under consideration by another Lévy process for which the double transform ${\mathbb E} e^{-\alpha \bar X_{\tau(q)}}$ is known, with τ(q) an exponentially distributed random variable with mean 1/q; then we use a fast and highly accurate Laplace inversion technique (of almost machine precision) to obtain the distribution of $\bar X_t$ . A broad range of examples illustrates the attractive features of our approach.

Suggested Citation

  • Naser M. Asghari & Peter Iseger & Michael Mandjes, 2014. "Numerical Techniques in Lévy Fluctuation Theory," Methodology and Computing in Applied Probability, Springer, vol. 16(1), pages 31-52, March.
    • Handle: RePEc:spr:metcap:v:16:y:2014:i:1:d:10.1007_s11009-012-9296-5
    DOI: 10.1007/s11009-012-9296-5
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    References listed on IDEAS

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    1. Carr, Peter, 1998. "Randomization and the American Put," The Review of Financial Studies, Society for Financial Studies, vol. 11(3), pages 597-626.
    2. Soeren Asmussen & Dilip Madan & Martijn Pistorius, 2007. "Pricing Equity Default Swaps under an approximation to the CGMY L\'{e}% vy Model," Papers 0711.2807, arXiv.org.
    3. Asmussen, Søren & Avram, Florin & Pistorius, Martijn R., 2004. "Russian and American put options under exponential phase-type Lévy models," Stochastic Processes and their Applications, Elsevier, vol. 109(1), pages 79-111, January.
    4. Joseph Abate & Ward Whitt, 1995. "Numerical Inversion of Laplace Transforms of Probability Distributions," INFORMS Journal on Computing, INFORMS, vol. 7(1), pages 36-43, February.
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    Cited by:

    1. O. J. Boxma & M. R. H. Mandjes, 2022. "Queueing and risk models with dependencies," Queueing Systems: Theory and Applications, Springer, vol. 102(1), pages 69-86, October.

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