Samoilenko I. V.
Ukr. Mat. Zh. - 2017. - 69, № 9. - pp. 1242-1249
The proposed methods enable us to study a model of stochastic evolution that includes Markov switchings and to identify the diffusion component and big jumps of perturbing process in the limiting equation. Big jumps of this type may describe rare catastrophic events in different applied problems. We consider the case where the perturbation of the system is determined by an impulse process in the nonclassical approximation scheme. Special attention is given to the asymptotic behavior of the generator of the analyzed evolutionary system.
Ukr. Mat. Zh. - 2015. - 67, № 8. - pp. 1151-1152
Ukr. Mat. Zh. - 2012. - 64, № 11. - pp. 1526-1535
Asymptotic analysis of the large deviation problem for impulsive processes in the scheme of Poisson approximation is performed. Large deviations for impulsive processes in the scheme of Poisson approximation are defined by an exponential generator for a jump process with independent increments.
Ukr. Mat. Zh. - 2010. - 62, № 5. - pp. 674–681
We propose an approach to the proof of the weak convergence of a semi-Markov process to a Markov process under certain conditions imposed on local characteristics of the semi-Markov process.
Ukr. Mat. Zh. - 2008. - 60, № 9. - pp. 1282–1286
We investigate an impulsive storage process switched by a jump process. The switching process is, in turn, averaged. We prove the weak convergence of the storage process in the scheme of series where a small parameter ε tends to zero.
Ukr. Mat. Zh. - 2006. - 58, № 9. - pp. 1234–1248
We determine the regular and singular components of the asymptotic expansion of a semi-Markov random evolution and show the regularity of boundary conditions. In addition, we propose an algorithm for finding initial conditions for t = 0 in explicit form using the boundary conditions for the singular component of the expansion.
Ukr. Mat. Zh. - 2002. - 54, № 3. - pp. 364-372
We introduce the notion of fading Markov random evolution and study the properties and characteristics of this process.
Ukr. Mat. Zh. - 2001. - 53, № 7. - pp. 1002-1008
We find moments of a process of Markov random evolutions in a finite-dimensional space.
A Probability Method for the Solution of the Telegraph Equation with Real-Analytic Initial Conditions
Ukr. Mat. Zh. - 2000. - 52, № 8. - pp. 1127-1134
We propose a method for the construction of an analytic solution of the Cauchy problem for the telegraph equation that is based on its simulation by a one-dimensional Markov random evolution.