Large deviations for impulsive processes in the scheme of the Lévy approximation
Author:
I. V. Samoĭlenko
Translated by:
N. Semenov
Journal:
Theor. Probability and Math. Statist. 88 (2014), 151-160
MSC (2010):
Primary 60J55, 60B10, 60F17, 60K10; Secondary 60G46, 60G60
DOI:
https://doi.org/10.1090/S0094-9000-2014-00925-4
Published electronically:
July 24, 2014
MathSciNet review:
3112641
Full-text PDF Free Access
Abstract |
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Additional Information
Abstract: Asymptotic analysis of the large deviations problem for impulsive processes in the scheme of the Lévy approximation is realized. Large deviations for impulsive processes in the scheme of the Lévy approximation are defined by the exponential generator for a jump process with independent increments.
References
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- Jin Feng and Thomas G. Kurtz, Large deviations for stochastic processes, Mathematical Surveys and Monographs, vol. 131, American Mathematical Society, Providence, RI, 2006. MR 2260560
- M. I. Freidlin and A. D. Wentzell, Random perturbations of dynamical systems, 2nd ed., Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 260, Springer-Verlag, New York, 1998. Translated from the 1979 Russian original by Joseph Szücs. MR 1652127
- Jean Jacod and Albert N. Shiryaev, Limit theorems for stochastic processes, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 288, Springer-Verlag, Berlin, 1987. MR 959133
- V. S. Korolyuk, Markov random evolutions with independent increments in an asymptotically small diffusion scheme, Dopov. Nats. Akad. Nauk Ukr. Mat. Prirodozn. Tekh. Nauki 6 (2010), 22–26 (Russian, with English and Ukrainian summaries). MR 3112727
- V. S. Korolyuk, Problem of large deviations for Markov random evolutions with independent increments in the scheme of asymptotically small diffusion, Ukraïn. Mat. Zh. 62 (2010), no. 5, 643–650 (Russian, with Russian summary); English transl., Ukrainian Math. J. 62 (2010), no. 5, 739–747. MR 2888630, DOI https://doi.org/10.1007/s11253-010-0384-9
- Vladimir S. Koroliuk and Nikolaos Limnios, Stochastic systems in merging phase space, World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2005. MR 2205562
- V. S. Koroliuk, N. Limnios, and I. V. Samoilenko, Lévy approximation of impulsive recurrent process with semi-Markov switching, Theory Stoch. Process. 16 (2010), no. 2, 77–85. MR 2777903
- V. S. Korolyuk, N. Līmnīos, and Ī. V. Samoĭlenko, The Lévy approximation of an impulsive recurrent process with Markov switchings, Teor. Ĭmovīr. Mat. Stat. 80 (2009), 15–22 (Ukrainian, with English, Russian and Ukrainian summaries); English transl., Theory Probab. Math. Statist. 80 (2010), 15–23. MR 2541948, DOI https://doi.org/10.1090/S0094-9000-2010-00791-5
- Ī. V. Samoĭlenko, Large deviations for random evolutions with independent increments in the scheme of Poisson approximation, Teor. Ĭmovīr. Mat. Stat. 85 (2011), 95–101 (Ukrainian, with English, Russian and Ukrainian summaries); English transl., Theory Probab. Math. Statist. 85 (2012), 107–114. MR 2933707, DOI https://doi.org/10.1090/S0094-9000-2013-00878-3
References
- S. N. Ethier and T. G. Kurtz, Markov Processes: Characterization and Convergence, J. Wiley & Sons, New York, 1986. MR 838085 (88a:60130)
- J. Feng and T. G. Kurtz, Large Deviation for Stochastic Processes, Mathematical Surveys and Monographs, vol. 131, American Mathematical Society, Providence, RI, 2006. MR 2260560 (2009g:60034)
- M. J. Freidlin and A. D. Wentzel, Random Perturbations of Dynamical Systems, Springer-Verlag, Berlin, 1998. MR 1652127 (99h:60128)
- J. Jacod and A. N. Shiryaev, Limit Theorems for Stochastic Processes, Springer-Verlag, Berlin, 1987. MR 959133 (89k:60044)
- V. S. Koroliuk, Markov random evolutions with independent increments in an asymptotically small diffusion scheme, Dopov. Nats. Akad. Nauk Ukr. Mat. Prirodozn. Tekh. Nauki (2010), no. 6, 22–26. (Ukrainian) MR 3112727
- V. S. Koroliuk, Problem of large deviations for Markov random evolutions with independent increments in the scheme of asymptotically small diffusion, Ukrain. Matem. Zh. 62 (2010), no. 5, 643–650; English transl. in Ukrainian Math. J. 62 (2010), no. 5, 739–747. MR 2888630
- V. S. Koroliuk and N. Limnios, Stochastic Systems in Merging Phase Space, World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2005. MR 2205562 (2007a:60004)
- V. S. Koroliuk, N. Limnios, and I. V. Samoĭlenko, Lévy approximation of impulsive recurrent process with semi-Markov switching, Theory Stoch. Process. 16(32) (2010), no. 2, 77–85. MR 2777903 (2011k:60169)
- V. S. Koroliuk, N. Limnios, and I. V. Samoĭlenko, The Lévy approximation of an impulsive recurrent process with Markov switchings, Teor. Imovir. Mat. Stat. 80 (2009), 85–92; English transl. in Theory Probab. Math. Statist. 80 (2010), 15–23. MR 2541948 (2010h:60101)
- I. V. Samoĭlenko, Large deviations for random evolutions with independent increments in the scheme of the Poisson approximation, Teor. Imovir. Mat. Stat. 85 (2011), 95–101; English transl. in Theory Probab. Math. Statist. 85 (2012), 107–114. MR 2933707
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Additional Information
I. V. Samoĭlenko
Affiliation:
Department of Fractal Analysis, Institute of Mathematics of National Academy of Science of Ukraine, Tereshchenkivs’ka Street, 3, Kyiv 01601, Ukraine
Email:
isamoil@imath.kiev.ua
Keywords:
Large deviations,
impulsive process,
Lévy approximation,
exponential nonlinear operator
Received by editor(s):
October 10, 2012
Published electronically:
July 24, 2014
Additional Notes:
The author is grateful to Academician of the National Academy of Science of Ukraine V. S. Koroliuk for posing the problem and for constant attention to its implementation
Article copyright:
© Copyright 2014
American Mathematical Society
