Laws of iterated logarithm for stochastic integrals of generalized sub-Gaussian processes
Authors:
A. Castellucci and R. Giuliano Antonini
Journal:
Theor. Probability and Math. Statist. 73 (2006), 47-56
MSC (2000):
Primary 60F15; Secondary 60G44
DOI:
https://doi.org/10.1090/S0094-9000-07-00680-1
Published electronically:
January 17, 2007
MathSciNet review:
2213840
Full-text PDF Free Access
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Abstract: We study the behavior of $\phi$-sub-Gaussian martingales $(M_t)_{t>0}$ as $t \to 0$. Applications are given to the stochastic integral of a particular kind of process and to the double stochastic integral of it with respect to two independent Brownian motions.
References
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- R. Giuliano Antonini and Yu. V. Kozachenko, A note on the asymptotic behavior of sequences of generalized subGaussian random vectors, Random Oper. Stochastic Equations 13 (2005), no. 1, 39–52. MR 2130246, DOI https://doi.org/10.1163/1569397053300900
- R. Giuliano Antonini, Yu. Kozachenko, and T. Nikitina, Spaces of $\phi $-sub-Gaussian random variables, Rend. Accad. Naz. Sci. XL Mem. Mat. Appl. (5) 27 (2003), 95–124 (English, with English and Italian summaries). MR 2056414
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References
- V. V. Buldygin and Yu. V. Kozachenko, Metric Characterization of Random Variables and Random Processes, AMS, Providence, RI, 2000. MR 1743716 (2001g:60089)
- W. Feller, An Introduction to Probability Theory and Its Applications, vol. II, Wiley, 1971. MR 0270403 (42:5292)
- R. Giuliano Antonini and Yu. V. Kozachenko, A note on the asymptotic behavior of sequences of generalized sub-Gaussian random vectors, Random Oper. Stochastic Equations 13 (2005), 39–52. MR 2130246 (2006b:60020)
- R. Giuliano Antonini, Yu. V. Kozachenko, and T. Nikitina, Spaces of $\phi$-sub-Gaussian random variables, Rend. Acc. Naz. delle Scienze detta dei XL, Mem. Mat. Appl. 121 (2003), 95–124. MR 2056414 (2005f:60036)
- K. Helmes, The local law of the iterated logarithm for processes related to Lévy’s stochastic area process, Studia Math. 83 (1986), 229–237. MR 850825 (87m:60096)
- P. Lévy, Wiener’s random function, and other Laplacian random functions, Proceedings of the second Berkeley Symposium on Mathematical Statistics and Probability, 1950. MR 0044774 (13:476b)
- R. Schott, Une Loi du Logarithme itéré pour certaines intégrales stochastiques, Comptes Rendus de l’Académie des Sciences Paris 292 (1981), 295–298. MR 609071 (82c:60142)
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Additional Information
A. Castellucci
Affiliation:
Dip. di Matematica, Università di Pisa, via F. Buonarroti 2, 56100 Pisa, Italy
Email:
castellucci@mail.dm.unipi.it
R. Giuliano Antonini
Affiliation:
Dip. di Matematica, Università di Pisa, via F. Buonarroti 2, 56100 Pisa, Italy
Email:
giuliano@dm.unipi.it
Keywords:
Continuous time martingale,
generalized sub-Gaussian process,
iterated logarithm law,
Brownian motion,
double stochastic integral,
Lévy area process
Received by editor(s):
July 30, 2004
Published electronically:
January 17, 2007
Article copyright:
© Copyright 2007
American Mathematical Society
