Law of the iterated logarithm for solutions of stochastic equations
Authors:
D. S. Budkov and S. Ya. Makhno
Translated by:
O. Klesov
Journal:
Theor. Probability and Math. Statist. 83 (2011), 47-57
MSC (2010):
Primary 60F10, 60F17
DOI:
https://doi.org/10.1090/S0094-9000-2012-00840-5
Published electronically:
February 2, 2012
MathSciNet review:
2768847
Full-text PDF Free Access
Abstract |
References |
Similar Articles |
Additional Information
Abstract: Strassen’s law of the iterated logarithm for a solution $x(t)$ of Itô’s stochastic equation is considered in the paper. We obtain a result for small times in the uniform metric and for a more general normalizing function than the classical $\sqrt { h\ln \ln \frac {1}{h}}$.
References
- A. V. Bulinskiĭ, A new variant of the functional law of the iterated logarithm, Teor. Veroyatnost. i Primenen. 25 (1980), no. 3, 502–512 (Russian, with English summary). MR 582580
- A. D. Wentzell, Limit theorems on large deviations for Markov stochastic processes, Mathematics and its Applications (Soviet Series), vol. 38, Kluwer Academic Publishers Group, Dordrecht, 1990. Translated from the Russian. MR 1135113
- M. I. Freidlin and A. D. Wentzell, Random perturbations of dynamical systems, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 260, Springer-Verlag, New York, 1984. Translated from the Russian by Joseph Szücs. MR 722136
- Nina Gantert, An inversion of Strassen’s law of the iterated logarithm for small time, Ann. Probab. 21 (1993), no. 2, 1045–1049. MR 1217579
- Davar Khoshnevisan, Exact rates of convergence to Brownian local time, Ann. Probab. 22 (1994), no. 3, 1295–1330. MR 1303646
- Dmitrii S. Budkov and Sergey Ya. Makhno, Functional iterated logarithm law for a Wiener process, Theory Stoch. Process. 13 (2007), no. 3, 22–28. MR 2396061
- I. I. Gikhman and A. V. Skorokhod, Stokhasticheskie differentsial′nye uravneniya i ikh prilozheniya, “Naukova Dumka”, Kiev, 1982 (Russian). MR 678374
References
- A. V. Bulinskiĭ, A new variant of the functional law of the iterated logarithm, Teor. Veroyatnost. Primenen. 25 (1980), 502–512; English transl. in Theory Probab. Appl. 25 (1981), 493–503. MR 582580 (82f:60066)
- A. D. Wentzell, Limit Theorems on Large Deviations for Markov Stochastic Processes, Nauka, Moscow, 1986; English transl., Kluwer Academic Publishers Group, Dordrecht, 1990. MR 1135113 (92i:60054)
- A. D. Wentzell and M. I. Freidlin, Random Perturbations of Dynamical Systems, Nauka, Moscow, 1979; English transl., Springer-Verlag, New York, 1984. MR 722136 (85a:60064)
- N. Gantert, An inversion of Strassen’s law of the iterated logarithm for small time, Ann. Probab. 21 (1993), 1045–1049. MR 1217579 (94d:60130)
- D. Khoshnevisan, Exact rates of convergence to Brownian local time, Ann. Probab. 22 (1994), 1295–1330. MR 1303646 (96b:60203)
- D. S. Budkov and S. Ya. Makhno, Functional iterated logarithm law for a Wiener process, Theory Stoch. Processes 13 (29) (2007), no. 3, 22–28. MR 2396061 (2008m:60159)
- I. I. Gikhman and A. V. Skorokhod, Stochastic Differential Equations and their Applications, Naukova dumka, Kiev, 1982. (Russian) MR 678374 (84j:60003)
Similar Articles
Retrieve articles in Theory of Probability and Mathematical Statistics
with MSC (2010):
60F10,
60F17
Retrieve articles in all journals
with MSC (2010):
60F10,
60F17
Additional Information
D. S. Budkov
Affiliation:
Department of Probability Theory and Mathematical Statistics, Institute of Applied Mathematics and Mechanics, National Academy of Sciences of Ukraine, R. Luxemburg Street 74, Donetsk 83114, Ukraine
Email:
budkov@iamm.ac.donetsk.ua
S. Ya. Makhno
Affiliation:
Department of Probability Theory and Mathematical Statistics, Institute of Applied Mathematics and Mechanics, National Academy of Sciences of Ukraine, R. Luxemburg Street 74, Donetsk 83114, Ukraine
Email:
makhno@iamm.ac.donetsk.ua
Keywords:
Stochastic equation,
large deviations,
law of the iterated logarithm
Received by editor(s):
December 10, 2009
Published electronically:
February 2, 2012
Additional Notes:
The research is supported by the Foundation for Joint Scientific Researches of National Academy of Science of Ukraine and Russian Foundation for Fundamental Researches, grant #104
Article copyright:
© Copyright 2012
American Mathematical Society
