We present a method of numerical approximation for stochastic integrals involving α-stable Lévy motion as an integrator. Constructions of approximate sums are based on the Poissonian series representation of such random measures. The main result gives an estimate of the rate of convergence of finite-dimensional distributions of finite sums approximating such stochastic integrals. Stochastic integrals driven by such measures are of interest in constructions of models for various problems arising in science and engineering, often providing a better description of real life phenomena than their Gaussian counterparts.
@article{bwmeta1.element.bwnjournal-article-zmv25i4p473bwm,
author = {Aleksander Janicki},
title = {Approximation of finite-dimensional distributions for integrals driven by $\alpha$-stable L\'evy motion},
journal = {Applicationes Mathematicae},
volume = {26},
year = {1999},
pages = {473-488},
zbl = {0998.60057},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-zmv25i4p473bwm}
}
Janicki, Aleksander. Approximation of finite-dimensional distributions for integrals driven by α-stable Lévy motion. Applicationes Mathematicae, Tome 26 (1999) pp. 473-488. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-zmv25i4p473bwm/
[000] L. Breiman (1968), (1992), Probability, 1st and 2nd eds., Addison-Wesley, Reading.
[001] 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.
[002] P. Embrechts, C. Klüppelberg and T. Mikosch (1997), Modelling Extremal Events, Springer, Berlin. | Zbl 0873.62116
[003] T. S. Ferguson and M. J. Klass (1972), A representation of independent increments processes without Gaussian components, Ann. Math. Statist. 43, 1634-1643. | Zbl 0254.60050
[004] A. Janicki and A. Weron (1994a), Simulation and Chaotic Behavior of α-Stable Stochastic Processes, Marcel Dekker, New York. | Zbl 0946.60028
[005] A. Janicki and A. Weron (1994b), Can one see α-stable variables and processes?, Statist. Sci. 9, 109-126. | Zbl 0955.60508
[006] 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. | Zbl 0567.60037
[007] T. Kurtz and P. Protter (1991), Weak limit theorems for stochastic integrals and stochastic differential equations, Ann. Probab. 19, 1035-1070. | Zbl 0742.60053
[008] M. Ledoux and M. Talagrand (1991), Probability Theory in Banach Spaces, Springer, Berlin. | Zbl 0748.60004
[009] R. LePage (1980), (1989), Multidimensional infinitely divisible variables and processes. Part I: Stable case, Technical Report No. 292, Department of Statistics, Stanford University; in: z Probability Theory on Vector Spaces IV (Łańcut, 1987), S. Cambanis and A. Weron (eds.), Lecture Notes in Math. 1391, Springer, New York, 153-163.
[010] W. Linde (1986), Probability in Banach Spaces-Stable and Infinitely Divisible Distributions, Wiley, New York. | Zbl 0665.60005
[011] K. R. Parthasarathy (1967), Probability Measures on Metric Spaces. Academic Press, New York and London. | Zbl 0153.19101
[012] P. Protter (1990), Stochastic Integration and Differential Equations-A New Approach, Springer, New York. | Zbl 0694.60047
[013] J. Rosinski (1990), On series representations of infinitely divisible random vectors, Ann. Probab. 18, 405-430. | Zbl 0701.60004
[014] G. Samorodnitsky and M. Taqqu (1994), Non-Gaussian Stable Processes: Stochastic Models with Infinite Variance, Chapman and Hall, London. | Zbl 0925.60027
[015] M. Shao and C. L. Nikias (1993), z Signal processing with fractional lower order moments: stable processes and their applications, Proc. IEEE 81, 986-1010.
[016] X. J. Wang (1992), Dynamical sporadicity and anomalous diffusion in the Lévy motion, Phys. Rev. A 45, 8407-8417.
[017] A. Weron (1984), z 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.