Self-similar models in risk theory

Krzysztof Burnecki

No HSC/98/03, HSC Research Reports from Hugo Steinhaus Center, Wroclaw University of Science and Technology

Abstract: 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.

Keywords: Self-similar process; Risk theory; Lamperti transformation; Insurance; Option pricing (search for similar items in EconPapers)
JEL-codes: C02 C46 G13 G22 (search for similar items in EconPapers)
Pages: 52 pages
Date: 1998
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