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## Applicationes Mathematicae

1996-1997 | 24 | 2 | 149-168
Tytuł artykułu

### Approximation of stochastic differential equations driven by α-stable Lévy motion

Autorzy
Treść / Zawartość
Warianty tytułu
Języki publikacji
EN
Abstrakty
EN
In this paper we present a result on convergence of approximate solutions of stochastic differential equations involving integrals with respect to α-stable Lévy motion. We prove an appropriate weak limit theorem, which does not follow from known results on stability properties of stochastic differential equations driven by semimartingales. It assures convergence in law in the Skorokhod topology of sequences of approximate solutions and justifies discrete time schemes applied in computer simulations. An example is included in order to demonstrate that stochastic differential equations with jumps are of interest in constructions of models for various problems arising in science and engineering, often providing better description of real life phenomena than their Gaussian counterparts. In order to demonstrate the usefulness of our approach, we present computer simulations of a continuous time α-stable model of cumulative gain in the Duffie-Harrison option pricing framework.
Słowa kluczowe
EN
Czasopismo
Rocznik
Tom
Numer
Strony
149-168
Opis fizyczny
Daty
wydano
1997
otrzymano
1995-09-18
poprawiono
1996-04-26
Twórcy
autor
• Hugo Steinhaus Center for Stochastic Methods, Technical University of Wrocław, 50-370 Wrocław, Poland
autor
• Hugo Steinhaus Center for Stochastic Methods, Technical University of Wrocław, 50-370 Wrocław, Poland
autor
• Hugo Steinhaus Center for Stochastic Methods, Technical University of Wrocław, 50-370 Wrocław, Poland
Bibliografia
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• D. Duffie and J. M. Harrison (1993), Arbitrage pricing of Russian options and perpetual lookback options, Ann. Appl. Probab. 3, 641-651.
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• A. Janicki and A. Weron (1994a), Simulation and Chaotic Behavior of α-Stable Stochastic Processes, Dekker, New York.
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• P. A. Kloeden, E. Platen and H. Schurz (1994), The Numerical Solution of SDE Through Computer Experiments, Springer, Berlin.
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• Z. Michna and I. Rychlik (1995), The expected number of level crossings for certain symmetric α-stable processes, Stochastic Models 11, 1-20.
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• A. Weron (1984), Stable processes and measures: A survey, in: Probability Theory on Vector Spaces III, D. Szynal and A. Weron (eds.), Lecture Notes in Math. 1080, Springer, New York, 306-364.
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