Large deviations problem for random evolution processes

Authors:
V. S. Koroliuk and I. V. Samoilenko

Translated by:
S. Kvasko

Original publication:
Teoriya Imovirnostei ta Matematichna Statistika, tom **93** (2015).

Journal:
Theor. Probability and Math. Statist. **93** (2016), 93-101

MSC (2010):
Primary 60J55, 60B10, 60F17, 60K10; Secondary 60G46, 60G60

DOI:
https://doi.org/10.1090/tpms/996

Published electronically:
February 7, 2017

Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: This is a short survey of the joint author results concerning the large deviations problem for some stochastic processes of random evolution published in the papers **[5]**-**[20]**.

**1.**S. N. Ethier and T. G. Kurtz,*Markov Processes: Characterization and Convergence*, Wiley, New York, 1985. MR**838085****2.**J. Feng,*Martingale problems for large deviations of Markov processes*, Stoch. Process. Appl.**81**(1999), 165-216. MR**1694569****3.**J. Feng and T. G. Kurtz,*Large Deviation for Stochastic Processes*, American Mathematical Society, RI, 2006. MR**2260560****4.**M. J. Freidlin and A. D. Wentzel,*Random Perturbations of Dynamical Systems*, Springer-Verlag, New York, 1998. MR**1652127****5.**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; English transl in Ukrainian Math. J.**62**(2010), no. 5, 739-747. MR**2888630****6.**V. S. Koroliuk,*Markov random evolutions with independent increments in the scheme of asymptotically small diffusion*, Dopovidi Nat. Akad. Sci. Ukraine (2010), no. 6, 22-26. (Russian) MR**3112727****7.**V. S. Koroliuk,*Large deviations problems for Markov random evolution with independent increments in the scheme of asymptotically small diffusion*, Comm. Statist. Theory Methods**40(19-20)**(2011), 3385-3395. MR**2860745****8.**V. S. Koroliuk,*Random evolutions with locally independent increments on increasing time intervals*, Ukr. Mat. Visn.**8**(2011), no. 2, 220-240; English transl. in J. Math. Sci.**179**(2011), no. 2, 273-289. MR**2883754****9.**V. S. Koroliuk,*Large deviations problem for random evolutions*, Visnyk Kyiv Taras Shevchenko University (2011), no. 25, 4-6. (Russian)**10.**V. S. Koroliuk,*Dynamic random evolutions on increasing time intervals*, Teor. Ĭmovir. Mat. Stat.**85**(2011), 75-83; English transl. in Theory Probab. Math. Statist.**85**(2012), 83-91. MR**2933705****11.**V. S. Koroliuk, R. Manca, and G. D'Amico,*Storage impulsive processes in the merging phase space*, Ukr. Mat. Visn.**10**(2013), no. 3, 333-342; J. Math. Sci.**196**(2014), no. 5, 644-651. MR**3253989****12.**V. S. Koroliuk, R. Manca, and G. D'Amico,*Storage impulsive processes on increasing time intervals*, Teor. Ĭmovir. Mat. Stat.**89**(2013), 64-74; English transl. in Theory Probab. Math. Statist.**89**(2014), 71-78. MR**3235176****13.**V. S. Koroliuk and I. V. Samoilenko,*Large deviations for impulse storage processes in the scheme of phase merging*, Dopovidi Nat. Akad. Sci. Ukraine (2014), no. 7, 28-35. (Ukrainian)**14.**V. S. Koroliuk and I. V. Samoilenko,*Large deviations for random evolutions in the scheme of asymptotically small diffusion*, Modern Stochastics and Applications, Springer Optimization and Its Applications**90**(2014), 201-217. MR**3236076****15.**I. V. Samoilenko,*Large deviations for impulsive processes in the scheme of Poisson approximation*, Ukraïn. Mat. Zh.**64**(2012), no. 11, 1526-1535. English transl. in Ukrainian Math. J.**64**(2013), no. 11, 1727-1738. MR**3104842****16.**I. V. Samoilenko,*Large deviations for random evolutions with independent increments in the scheme of Poisson approximation*, Teor. Ĭmovir. Mat. Stat. no. 85 (2011), 95-101; English transl. in Theory Probab. Math. Statist.**85**(2012), 107-114. MR**2933707****17.**I. V. Samoilenko,*Large deviations for impulsive processes in the scheme of the Lévy approximation*, Teor. Ĭmovir. Mat. Stat. no. 88 (2014), 135-143; English transl. in Theory Probab. Math. Statist.**88**(2015), 151-160. MR**3112641****18.**I. V. Samoilenko and Yu. V. Shusharin,*Large deviations for a random evolution with independent increments in the scheme of the Poisson approximation with split and double merging*, Zh. Obchysl. Prykl. Mat.**117**(2014), no. 3, 76-86. (Ukrainian) MR**3353628****19.**I. V. Samoilenko,*Large deviations for random evolutions with independent increments in the scheme of Lévy approximation with split and double merging*, Random Oper. Stoch. Equ.**23**(2015), no. 2, 137-149. MR**3353628****20.**I. V. Samoilenko,*Large deviations for random evolutions with independent increments in a scheme of Lévy approximation*, Ukr. Mat. Visn.**12**(2015), no. 1, 67-85; English transl. in J. Math. Sci.**210**(2015), no. 1, 52-66. MR**3400243**

Retrieve articles in *Theory of Probability and Mathematical Statistics*
with MSC (2010):
60J55,
60B10,
60F17,
60K10,
60G46,
60G60

Retrieve articles in all journals with MSC (2010): 60J55, 60B10, 60F17, 60K10, 60G46, 60G60

Additional Information

**V. S. Koroliuk**

Affiliation:
Institute of Mathematics, National Academy of Science of Ukraine, Tereshchenkivs’ka Street, 3, Kyiv 01601, Ukraine

Email:
vskorol@yahoo.com

**I. V. Samoilenko**

Affiliation:
Faculty for Computer Science and Cybernetics, National Taras Shevchenko University, Academician Glushkov Avenue, 4, Kyiv 03127, Ukraine

Email:
isamoil@i.ua

DOI:
https://doi.org/10.1090/tpms/996

Keywords:
Large deviations,
random evolution processes,
asymptotically small diffusion,
L\'evy approximation,
exponential nonlinear generator,
split and double merging

Received by editor(s):
June 29, 2015

Published electronically:
February 7, 2017

Additional Notes:
This paper was prepared following the talk at the International conference “Probability, Reliability and Stochastic Optimization (PRESTO-2015)” held in Kyiv, Ukraine, April 7–10, 2015

Article copyright:
© Copyright 2017
American Mathematical Society