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Self-similar models in risk theory


  • Krzysztof Burnecki


This Ph.D. thesis is concerned with self-similar processes. In Chapter 2 we describe the classes of transformations leading from self-similar to stationary processes, and conversely. The relationship is used in Chapter 3 to characterize stable symmetric self-similar processes via their minimal integral representation. This leads to a unique decomposition of a symmetric stable self-similar process into three independent parts. The class of such processes appears to be quite broad and can stand as a basis of different risk models. In Chapter 4 we give examples of applications of self-similar processes in insurance risk modelling. In Chapter 5 we illustrate a test of self-similarity (namely variance-time plots) on DJIA index data in order to justify the use of self-similar processes in financial modelling. Last but not least we propose an alternative model for stock price movements incorporating a martingale which generates the same filtration as fractional Brownian motion.

Suggested Citation

  • Krzysztof Burnecki, 1998. "Self-similar models in risk theory," HSC Research Reports HSC/98/03, Hugo Steinhaus Center, Wroclaw University of Technology.
  • Handle: RePEc:wuu:wpaper:hsc9803

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    File Function: Final version, 25 May 1998
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    References listed on IDEAS

    1. Aleksander Janicki & Aleksander Weron, 1994. "Simulation and Chaotic Behavior of Alpha-stable Stochastic Processes," HSC Books, Hugo Steinhaus Center, Wroclaw University of Technology, number hsbook9401, June.
    2. Krzysztof Burnecki & Makoto Maejima & Aleksander Weron, 1997. "The Lamperti transformation for self-similar processes," HSC Research Reports HSC/97/02, Hugo Steinhaus Center, Wroclaw University of Technology.
    3. Weron, Aleksander & Mercik, Szymon & Weron, Rafal, 1999. "Origins of the scaling behaviour in the dynamics of financial data," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 264(3), pages 562-569.
    4. L. C. G. Rogers, 1997. "Arbitrage with Fractional Brownian Motion," Mathematical Finance, Wiley Blackwell, vol. 7(1), pages 95-105.
    5. Krzysztof Burnecki & Jan Rosinski & Aleksander Weron, 1997. "Spectral representation and structure of self-similar processes," HSC Research Reports HSC/97/03, Hugo Steinhaus Center, Wroclaw University of Technology.
    6. Furrer, Hansjorg & Michna, Zbigniew & Weron, Aleksander, 1997. "Stable Lévy motion approximation in collective risk theory," Insurance: Mathematics and Economics, Elsevier, vol. 20(2), pages 97-114, September.
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    More about this item


    Self-similar process; Risk theory; Lamperti transformation; Insurance; Option pricing;

    JEL classification:

    • C02 - Mathematical and Quantitative Methods - - General - - - Mathematical Economics
    • C46 - Mathematical and Quantitative Methods - - Econometric and Statistical Methods: Special Topics - - - Specific Distributions
    • G13 - Financial Economics - - General Financial Markets - - - Contingent Pricing; Futures Pricing
    • G22 - Financial Economics - - Financial Institutions and Services - - - Insurance; Insurance Companies; Actuarial Studies


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