# Bondarev B. V.

### Intermixing “according to Ibragimov”. Estimate for rate of approach of family of integral functionals of solution of differential equation with periodic coefficients to family of the Wiener processes. Some applications. II

Ukr. Mat. Zh. - 2011. - 63, № 3. - pp. 303-318

In the first part of this work, we obtain estimates for the rate of approach of integrals of a family of "physical" white noises to a family of the Wiener processes. By using this result, we establish an estimate for the rate of approach of a family of solutions of ordinary differential equations, disturbed by some physical white noises, to a family of solutions of the corresponding Ito equations. We consider the case where the coefficient of random disturbance is separated from zero as well as the case where it is not separated from zero.

### Mixing “In the sense of Ibragimov.” Estimate for the rate of approach of a family of integral functionals of a solution of a differential equation with periodic coefficients to a family of wiener processes. Some applications. I

Ukr. Mat. Zh. - 2010. - 62, № 6. - pp. 733–753

We prove that a bounded 1-periodic function of a solution of a time-homogeneous diffusion equation with 1-periodic coefficients forms a process that satisfies the condition of uniform strong mixing. We obtain an estimate for the rate of approach of a certain normalized integral functional of a solution of an ordinary time-homogeneous stochastic differential equation with 1-periodic coefficients to a family of Wiener processes in probability in the metric of space $C[0, T]$. As an example, we consider an ordinary differential equation perturbed by a rapidly oscillating centered process that is a 1-periodic function of a solution of a time-homogeneous stochastic differential equation with 1-periodic coefficients. We obtain an estimate for the rate of approach of a solution of this equation to a solution of the corresponding Itô stochastic equation.

### On the ε-sufficient control in one merton problem with “physical” white noise

Ukr. Mat. Zh. - 2009. - 61, № 8. - pp. 1025-1039

We consider the Merton problem of finding the strategies of investment and consumption in the case where the evolution of risk assets is described by the exponential model and the role of the main process is played by the integral of a certain stationary “physical” white noise generated by the centered Poisson process. It is shown that the optimal controls computed for the limiting case are ε-sufficient controls for the original system.

### Evaluation of the probability of bankruptcy for a model of insurance company

Bondarev B. V., Zhmykhova T. V.

Ukr. Mat. Zh. - 2007. - 59, № 4. - pp. 447–457

A problem of calculating the probability of ruin of an insurance company in infinite number of steps is considered in the case where this company is able to invest its capital to a bank deposit at every time. As a distribution describing claim amounts to the insurance company, the gamma distribution with parameters $n$ and $\alpha$ is chosen.

### Order reduction for a system of stochastic differential equations with a small parameter in the coefficient of the leading derivative. Estimate for the rate of convergence

Ukr. Mat. Zh. - 2006. - 58, № 12. - pp. 1587–1601

In the metric $\rho(X, Y) = (\sup\limits_{0 \leq t \leq T} M|X(t) - Y(t)|^2)^{1/2} $ for an ordinary stochastic differential equation of order $p \geq 2$ with small parameter of the higher derivative, we establish an estimate of the rate of convergence of its solution to a solution of stochastic equation of order $p - 1$.

### Invariance principle for one class of Markov chains with fast Poisson time. Estimate for the rate of convergence

Ukr. Mat. Zh. - 2006. - 58, № 9. - pp. 1155–1174

We obtain an estimate for the rate of convergence of normalized Poisson sums of random variables determined by the first-order autoregression procedure to a family of Wiener processes.

### A Stochastic Analog of Bogolyubov's Second Theorem

Ukr. Mat. Zh. - 2005. - 57, № 7. - pp. 879–894

We establish an estimate for the rate at which a solution of an ordinary differential equation subject to the action of an ergodic random process converges to a stationary solution of a deterministic averaged system on time intervals of order $e^{1/ερ}$ for some $0 < ρ < 1$.

### Estimates for the Rate of Convergence in Ordinary Differential Equations under the Action of Random Processes with Fast Time

Ukr. Mat. Zh. - 2005. - 57, № 4. - pp. 435–457

We study the procedure of averaging in the Cauchy problem for an ordinary differential equation perturbed by a certain Markov ergodic process. We establish several estimates for the rate of convergence of solutions of the original problem to solutions of the averaged one.

### Estimation of an Unknown Parameter in the Cauchy Problem for a First-Order Partial Differential Equation under Small Gaussian Perturbations

Bondarev B. V., Dzundza A. I., Simogin A. A.

Ukr. Mat. Zh. - 2000. - 52, № 7. - pp. 999-1006

On the basis of observation of a realization of a solution of the Cauchy problem, we establish a maximum-likelihood estimate for an unknown parameter. We construct an exponential inequality for the probabilities of large deviations of the estimate from the real value of the parameter.

### Functional law of the iterated logarithm for fields and its applications

Ukr. Mat. Zh. - 1997. - 49, № 7. - pp. 883–894

For a Wiener field with an arbitrary finite number of parameters, we construct the law of the iterated logarithm in the functional form. We consider the problem for random fields of a certain type to reside within curvilinear boundaries without assuming that the Cairoli—Walsh condition is satisfied.

### Functional law of iterated logarithm for normalized integrals of processes with weak dependence

Ukr. Mat. Zh. - 1997. - 49, № 4. - pp. 490–499

For normalized integrals of processes with weak dependence, we prove the law of iterated logarithm in the Strassen form. The results obtained are used for the construction of a curvilinear confidence region in which a solution of an equation with small parameter is sought.

### Оценка рассчетных воздействий в параболических системах. $L_2$-подход

Ukr. Mat. Zh. - 1996. - 48, № 1. - pp. 3-12

By using observations of solutions of the first initial boundary-value problem for a parabolic quasilinear equation with fast random oscillations, we estimate the nonlinear term of the equation. In the metric of the space $L_2$, we study large deviations of a nonparametric estimate of nonlinear influence.

### Averaging in hyperbolic systems subject to weakly dependent random perturbations

Ukr. Mat. Zh. - 1992. - 44, № 8. - pp. 1011–1020

The first initial boundary value problem is considered for a hyperbolic equation with a small parameter for an external action described by some stochastic process satisfying some of the conditions of weak dependence. Averaging of the coefficients over the temporal variable is conducted. The existence is assumed of a unique generalized solution both for the initial stochastic problem and for the problem with an “averaged” equation, which turns out to be deterministic. For the probability of deviation of a solution of the initial equation from the solution of the “averaged” problem, exponential bounds are established of the type of S. N. Bernshtein inequalities for the sums of independent random variables.

### Averaging in parabolic systems, subjected to weakly dependent random perturbations. The *L*_{1}-approach

Ukr. Mat. Zh. - 1991. - 43, № 2. - pp. 167-172

### Estimates of probabilities of large deviations in problems of estimation of rated effects. I

Ukr. Mat. Zh. - 1991. - 43, № 1. - pp. 27–35

### Averaging stochastic systems under weakly dependent perturbations

Ukr. Mat. Zh. - 1990. - 42, № 5. - pp. 593–600

### Averagings in stochastic systems with dependence on the whole past

Ukr. Mat. Zh. - 1990. - 42, № 4. - pp. 443–451

### Exponential bounds in stochastic approximation procedures

Ukr. Mat. Zh. - 1989. - 41, № 7. - pp. 867–872

### Averages in curvilinear boundaries of stochastic hyperbolic systems

Bondarev B. V., Vorob'eva I. L.

Ukr. Mat. Zh. - 1989. - 41, № 6. - pp. 828-831

### Kolmogorov statistic in the case of a piecewise-continuous distribution function

Ukr. Mat. Zh. - 1988. - 40, № 2. - pp. 145-149