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

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

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.
Rocznik
Tom
24
Numer
2
Strony
149-168
Opis fizyczny
Daty
wydano
1997
otrzymano
1995-09-18
poprawiono
1996-04-26
Twórcy
  • Hugo Steinhaus Center for Stochastic Methods, Technical University of Wrocław, 50-370 Wrocław, Poland
  • Hugo Steinhaus Center for Stochastic Methods, Technical University of Wrocław, 50-370 Wrocław, Poland
  • Hugo Steinhaus Center for Stochastic Methods, Technical University of Wrocław, 50-370 Wrocław, Poland
Bibliografia
  • R. J. Adler, G. Samorodnitsky and T. Gadrich (1993), The expected number of level crossings for stationary, harmonisable, symmetric, stable processes, Ann. Appl. Probab. 3, 553-575.
  • P. Billingsley (1968), Convergence of Probability Measures, Wiley, New York.
  • S. V. Buldyrev, A. L. Goldberger, S. Havlin, C.-K. Peng, M. Simons and H. E. Stanley (1993), Generalized Lévy walk model for DNA nucleotide sequences, Phys. Rev. E 47, 4514-4523.
  • D. Duffie and J. M. Harrison (1993), Arbitrage pricing of Russian options and perpetual lookback options, Ann. Appl. Probab. 3, 641-651.
  • P. Embrechts and H. Schmidli (1994), Modelling of extremal events in insurance and finance, Math. Methods Oper. Res. 39, 1-34.
  • S. N. Ethier and T. G. Kurtz (1986), Markov Processes: Characterization and Convergence, Wiley, New York.
  • W. Feller (1971), An Introduction to Probability Theory and its Applications, Vol. 2, Wiley, New York.
  • N. Ikeda and S. Watanabe (1981), Stochastic Differential Equations and Diffusion Processes. North-Holland, Amsterdam.
  • P. Jagers (1974), Aspect of random measures and point processes, in: Advances in Probability and Related Topics, Vol. 3, Dekker, New York, 1974, 306-364.
  • A. Jakubowski, J. Mémin et G. Pages (1989), Convergence en loi des suites d'intégrales stochastiques sur l'espace ${\sym D}^1$ de Skorokhod, Probab. Theory Related Fields 81, 111-137.
  • A. Janicki and A. Weron (1994a), Simulation and Chaotic Behavior of α-Stable Stochastic Processes, Dekker, New York.
  • A. Janicki and A. Weron (1994b), Can one see α-stable variables and processes? Statist. Sci. 9, 109-126.
  • I. Karatzas and S. E. Shreve (1988), Brownian Motion and Stochastic Calculus, Springer, New York.
  • Y. Kasahara and M. Maejima (1986), Functional limit theorems for weighted sums of i.i.d. random variables, Probab. Theory Related Fields 72, 161-183.
  • Y. Kasahara and M. Maejima (1988), Weighted sums of i.i.i. random variables attracted to integrals of stable processes, ibid. 78, 75-96.
  • Y. Kasahara and S. Watanabe (1986), Limit theorems for point processes and their functionals, J. Math. Soc. Japan 38, 543-574.
  • Y. Kasahara and K. Yamada (1991), Stability theorem for stochastic differential equations with jumps, Stochastic Process. Appl. 38, 13-32.
  • O. Kella (1993), Parallel and tandem fluid networks with dependent Lévy inputs, Ann. Appl. Probab. 3, 682-695.
  • H. Kesten and G. C. Papanicolaou (1979), A limit theorem for turbulent diffusion, Comm. Math. Phys. 65, 97-128.
  • P. A. Kloeden and E. Platen (1992), The Numerical Solution of Stochastic Differential Equations, Springer, Heidelberg.
  • P. A. Kloeden, E. Platen and H. Schurz (1994), The Numerical Solution of SDE Through Computer Experiments, Springer, Berlin.
  • T. G. Kurtz and P. Protter (1991), Weak limit theorems for stochastic integrals and stochastic differential equations, Ann. Probab. 19, 1035-1070.
  • Z. Michna and I. Rychlik (1995), The expected number of level crossings for certain symmetric α-stable processes, Stochastic Models 11, 1-20.
  • E. Pardoux and D. Talay (1985), Discretization and simulation of stochastic differential equations, Acta Appl. Math. 3, 23-47.
  • K. R. Parthasarathy (1967), Probability Measures on Metric Spaces, Academic Press, New York and London.
  • P. Protter (1990), Stochastic Integration and Differential Equations-A New Approach, Springer, New York.
  • S. T. Rachev and G. Samorodnitsky (1993), Option pricing formula for speculative prices modelled by subordinated stochastic processes, Serdica 19, 175-190.
  • S. I. Resnick (1987), Extreme Values, Regular Variation, and Point Processes, Springer, New York.
  • G. Samorodnitsky and M. Taqqu (1994), Non-Gaussian Stable Processes: Stochastic Models with Infinite Variance, Chapman & Hall, London.
  • M. Shao and C. L. Nikias (1993), Signal processing with fractional lower order moments: stable processes and their applications, Proc. IEEE 81, 986-1010.
  • L. Słomiński (1989), Stability of strong solutions of stochastic differential equations, Stochasic Process. Appl. 31, 173-202.
  • X. J. Wang (1992), Dynamical sporadicity and anomalous diffusion in the Lévy motion, Phys. Rev. A 45, 8407-8417.
  • 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.
Typ dokumentu
Bibliografia
Identyfikatory
Identyfikator YADDA
bwmeta1.element.bwnjournal-article-zmv24i2p149bwm
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