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Theory of Probability and Mathematical Statistics



Large deviations problem for random evolution processes

V. S. Koroliuk, I. V. Samoilenko

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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].

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

Bibliography:
1. S. N. Ethier and T. G. Kurtz, Markov Processes: Characterization and Convergence, Wiley, New York, 1985.
2. J. Feng, Martingale problems for large deviations of Markov processes, Stoch. Process. Appl. 81 (1999), 165-216.
3. J. Feng and T. G. Kurtz, Large Deviation for Stochastic Processes, American Mathematical Society, RI, 2006.
4. M. J. Freidlin and A. D. Wentzel, Random Perturbations of Dynamical Systems, Springer-Verlag, New York, 1998.
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.
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)
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.
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.
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.
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.
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.
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.
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.
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.
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.
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)
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.
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.