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Can One See Alpha-stable Variables and Processes?

Author

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  • Aleksander Janicki
  • Aleksander Weron

Abstract

In this paper, we demonstrate some properties of Alpha-stable (stable) random variables and processes. It turns out that with the use of suitable statistical estimation techniques, computer simulation procedures and numerical discretization methods it is possible to construct approximations of stochastic integrals with stable measures as integrators. As a consequence we obtain an effective, general method giving approximate solutions for a wide class of stochastic differential equations involving such integrals. Application of computer graphics provides interesting quantitative and visual information on those features of stable variates which distinguish them from their commonly used Gaussian counterparts. It is possible to demonstrate evolution in time of densities with heavy tails of appropriate processes, to visualize the effect of jumps of trajectories, etc. We try to demonstrate that stable variates can be very useful in stochastic modeling of problems of different kinds, arising in science and engineering, which often provide better description of real life phenomena than their Gaussian counterparts.

Suggested Citation

  • Aleksander Janicki & Aleksander Weron, 1994. "Can One See Alpha-stable Variables and Processes?," HSC Research Reports HSC/94/01, Hugo Steinhaus Center, Wroclaw University of Science and Technology.
    • Handle: RePEc:wuu:wpaper:hsc9401
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    Citations

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    Cited by:

    1. Aleksander Janicki & Zbigniew Michna & Aleksander Weron, 1996. "Approximation of stochastic differential equations driven by alpha-stable Levy motion," HSC Research Reports HSC/96/02, Hugo Steinhaus Center, Wroclaw University of Science and Technology.
    2. Adam Misiorek & Rafal Weron, 2010. "Heavy-tailed distributions in VaR calculations," HSC Research Reports HSC/10/05, Hugo Steinhaus Center, Wroclaw University of Science and Technology.
    3. Borak, Szymon & Misiorek, Adam & Weron, Rafał, 2010. "Models for heavy-tailed asset returns," SFB 649 Discussion Papers 2010-049, Humboldt University Berlin, Collaborative Research Center 649: Economic Risk.
    4. Grzegorz Krzy.zanowski & Marcin Magdziarz, 2020. "A computational weighted finite difference method for American and barrier options in subdiffusive Black-Scholes model," Papers 2003.05358, arXiv.org, revised Dec 2020.
    5. Janicki, Aleksander & Weron, Aleksander, 1995. "Computer simulation of attractors in stochastic models with α-stable noise," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 39(1), pages 9-19.
    6. Liu, He & Song, Wanqing & Li, Ming & Kudreyko, Aleksey & Zio, Enrico, 2020. "Fractional Lévy stable motion: Finite difference iterative forecasting model," Chaos, Solitons & Fractals, Elsevier, vol. 133(C).
    7. Weron, Rafał, 2004. "Computationally intensive Value at Risk calculations," Papers 2004,32, Humboldt University of Berlin, Center for Applied Statistics and Economics (CASE).
    8. Shibin Zhang, 2011. "Transition Law-based Simulation of Generalized Inverse Gaussian Ornstein–Uhlenbeck Processes," Methodology and Computing in Applied Probability, Springer, vol. 13(3), pages 619-656, September.

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