2019
Том 71
№ 1

# Chabanyuk Ya. M.

Articles: 6
Article (Ukrainian)

### Procedure of stochastic approximation for the diffusion process with semi-Markov switchings

Ukr. Mat. Zh. - 2018. - 70, № 11. - pp. 1563-1570

We obtain sufficient conditions for the convergence of the procedure of stochastic approximation for the diffusion process in the case of a uniformly ergodic semi-Markov process of switchings of the regression function with the use of a small parameter in the scheme of series.

Article (Ukrainian)

### Stability of a dynamical system with semi-Markov switchings under conditions of diffusion approximation

Ukr. Mat. Zh. - 2007. - 59, № 9. - pp. 1290–1296

We obtain sufficient conditions for the stability of a dynamical system in a semi-Markov medium under the conditions of diffusion approximation by using asymptotic properties of the compensation operator for a semi-Markov process and properties of the Lyapunov function for an averaged system.

Article (Ukrainian)

### Asymptotic normality of fluctuations of the procedure of stochastic approximation with diffusive perturbation in a Markov medium

Ukr. Mat. Zh. - 2006. - 58, № 12. - pp. 1686–1692

We consider the asymptotic normality of a continuous procedure of stochastic approximation in the case where the regression function contains a singularly perturbed term depending on the external medium described by a uniformly ergodic Markov process. Within the framework of the scheme of diffusion approximation, we formulate sufficient conditions for asymptotic normality in terms of the existence of a Lyapunov function for the corresponding averaged equation.

Article (Ukrainian)

### Asymptotic normality of a discrete procedure of stochastic approximation in a semi-Markov medium

Ukr. Mat. Zh. - 2006. - 58, № 10. - pp. 1425–1433

We obtain sufficient conditions for the asymptotic normality of a jump procedure of stochastic approximation in a semi-Markov medium using a compensating operator of an extended Markov renewal process. The asymptotic representation of the compensating operator guarantees the construction of the generator of a limit diffusion process of the Ornstein-Uhlenbeck type.

Brief Communications (Ukrainian)

### Continuous procedure of stochastic approximation in a semi-Markov medium

Ukr. Mat. Zh. - 2004. - 56, № 5. - pp. 713–720

Using the Lyapunov function for an averaged system, we establish conditions for the convergence of the procedure of stochastic approximation $$du(t)=a(t)[C(u(t),x(t))dt+σ(u(t))dw(t)]$$ in a random semi-Markov medium described by an ergodic semi-Markov process $x(t)$.

Article (Ukrainian)

### Stability of a Dynamical System with Semi-Markov Switchings under Conditions of Stability of the Averaged System

Ukr. Mat. Zh. - 2002. - 54, № 2. - pp. 195-204

We establish additional stability conditions on the rate of a dynamical system with semi-Markov switchings and on the Lyapunov function for the averaged system.