Yuriy Kozachenko

On drift parameter estimation in models with fractional Brownian motion

We consider a stochastic differential equation involving standard and fractional Brownian motion with unknown drift parameter to be estimated. We investigate the standard maximum likelihood estimate of the drift parameter, two non-standard estimates and three estimates for the sequential estimation. Model strong consistency and some other properties are proved. The linear model and Ornstein–Uhlenbeck model are studied in detail. As an auxiliary result, an asymptotic behaviour of the fractional derivative of the fractional Brownian motion is established.

Uniform Convergence of Wavelet Expansions of Gaussian Random Processes

New results on uniform convergence in probability for the most general classes of wavelet expansions of stationary Gaussian random processes are given.

Upper estimate of overrunning by Sub φ (Ω) random process the level specified by continuous function

In this paper we consider random process from the space Sub φ (Ω), which is defined on compact set, and the probability that supremum of this process exceeds some function. The class of Sub φ (Ω) random processes is more general than the class of Gaussian processes. By applying obtained estimation to a fluid queue fed by a process of Ornstein-Uhlenbeck from the space strictly Sub φ (Ω), where we show that for interval [a, b] there exist constants A, B, D and for large enough buffer capacity x.

Asymptotic growth of trajectories of multifractional Brownian motion, with statistical applications to drift parameter estimation

We construct the least-square estimator for the unknown drift parameter in the multifractional Ornstein–Uhlenbeck model and establish its strong consistency in the non-ergodic case. The proofs are based on the asymptotic bounds with probability 1 for the rate of the growth of the trajectories of multifractional Brownian motion (mBm) and of some other functionals of mBm, including increments and fractional derivatives. As the auxiliary results having independent interest, we produce the asymptotic bounds with probability 1 for the rate of the growth of the trajectories of the general Gaussian process and some functionals of it, in terms of the covariance function of its increments.

Uniform Convergence of Compactly Supported Wavelet Expansions of Gaussian Random Processes

New results on uniform convergence in probability for expansions of Gaussian random processes using compactly supported wavelets are given. The main result is valid for general classes of non stationary processes. An application of the obtained results to stationary processes is also presented. It is shown that the convergence rate of the expansions is exponential.

Convergence Rate of Wavelet Expansions of Gaussian Random Processes

This article characterizes uniform convergence rate for general classes of wavelet expansions of stationary Gaussian random processes. The convergence in probability is considered.

The criterion of hypothesis testing on the covariance function of a Gaussian stochastic process

Abstract. We consider a square Gaussian stochastic process. Estimates of the distribution of some functional of this process are obtained. Tests for a hypothesis concerning the form of the covariance function of a Gaussian stochastic process are constructed.

Whittaker–Kotel'nikov–Shannon approximation of φ-sub-Gaussian random processes

Abstract The article starts with generalizations of some classical results and new truncation error upper bounds in the sampling theorem for bandlimited stochastic processes. Then, it investigates L p ( [ 0 , T ] ) and uniform approximations of φ-sub-Gaussian random processes by finite time sampling sums. Explicit truncation error upper bounds are established. Some specifications of the general results for which the assumptions can be easily verified are given. Direct analytical methods are employed to obtain the results.

A criterion for testing hypotheses about the covariance function of a stationary Gaussian stochastic process

We consider a measurable stationary Gaussian stochastic process. A criterion for testing hypotheses about the covariance function of such a process using estimates for its norm in the space

The Cauchy problem for the heat equation with a random right side

L_p(\mathbb {T}),\,p\geq1