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Kostiantyn Ralchenko

Publications and source records attributed to Kostiantyn Ralchenko.

At least 37 records · Page 2Linked to original sources
arXiv2022

Analytical and computational problems related to fractional Gaussian noise

We study the projection of an element of fractional Gaussian noise onto its neighbouring elements. We prove some analytic results for the coefficients of this projection, in particular, we obtain recurrence relations for them. We also make several conjectures concerning the behaviour of these coefficients, provide numerical evidence supporting these conjectures, and study them theoretically in particular cases. As an auxiliary result of independent interest, we investigate the covariance function of fractional Gaussian noise, prove that it is completely monotone for $H>1/2$, and, in particular monotone, convex, log-convex along with further useful properties.

math.PR↗
arXiv2021

Parameter estimation in CKLS model by continuous observations

We consider a stochastic differential equation of the form $dr_t = (a - b r_t) dt + σr_t^βdW_t$, where $a$, $b$ and $σ$ are positive constants, $β\in(\frac12,1)$. We study the estimation of an unknown drift parameter $(a,b)$ by continuous observations of a sample path $\{r_t, t \in [0,T]\}$. We prove the strong consistency and asymptotic normality of the maximum likelihood estimator. We propose another strongly consistent estimator, which generalizes an estimator proposed in Dehtiar et al. (2021) for $β=\frac12$. The identification of the diffusion parameters $σ$ and $β$ is discussed as well.

math.ST↗
arXiv2020

Stochastic differential equations driven by additive Volterra-Lévy and Volterra-Gaussian noises

We study the existence and uniqueness of solutions to stochastic differential equations with Volterra processes driven by Lévy noise. For this purpose, we study in detail smoothness properties of these processes. Special attention is given to two kinds of Volterra-Gaussian processes that generalize the compact interval representation of fractional Brownian motion and to stochastic equations with such processes.

math.PR↗
arXiv2020

Two methods of estimation of the drift parameters of the Cox-Ingersoll-Ross process: continuous observations

We consider a stochastic differential equation of the form $dr_t = (a - b r_t) dt + σ\sqrt{r_t}dW_t$, where $a$, $b$ and $σ$ are positive constants. The solution corresponds to the Cox-Ingersoll-Ross process. We study the estimation of an unknown drift parameter $(a,b)$ by continuous observations of a sample path $\{r_t,t\in[0,T]\}$. First, we prove the strong consistency of the maximum likelihood estimator. Since this estimator is well-defined only in the case $2a>σ^2$, we propose another estimator that is defined and strongly consistent for all positive $a$, $b$, $σ$. The quality of the estimators is illustrated by simulation results.

math.PR↗
arXiv2020

Existence and uniqueness of mild solution to fractional stochastic heat equation

For a class of non-autonomous parabolic stochastic partial differential equations defined on a bounded open subset $D\subset \mathbb {R}^d$ and driven by an $L^2(D)$-valued fractional Brownian motion with the Hurst index $H>1/2$, a new result on existence and uniqueness of a mild solution is established. Compared to the existing results, the uniqueness in a fully nonlinear case is shown, not assuming the coefficient in front of the noise to be affine. Additionally, the existence of moments for the solution is established.

math.PR↗
arXiv2020

Maximum likelihood estimation in the non-ergodic fractional Vasicek model

We investigate the fractional Vasicek model described by the stochastic differential equation $dX_t=(α-βX_t)\,dt+γ\,dB^H_t$, $X_0=x_0$, driven by the fractional Brownian motion $B^H$ with the known Hurst parameter $H\in (1/2,1)$. We study the maximum likelihood estimators for unknown parameters $α$ and $β$ in the non-ergodic case (when $β<0$) for arbitrary $x_0\in \mathbb{R}$, generalizing the result of Tanaka, Xiao and Yu (2019) for particular $x_0=α/β$, derive their asymptotic distributions and prove their asymptotic independence.

math.PR↗
arXiv2019

Fractional stochastic heat equation with piecewise constant coefficients

We introduce a fractional stochastic heat equation with second order elliptic operator in divergence form, having a piecewise constant diffusion coefficient, and driven by an infinite-dimensional fractional Brownian motion. We characterize the fundamental solution of its deterministic part, and prove the existence and the uniqueness of its solution.

math.PR↗
arXiv2018

Parameter estimation for Gaussian processes with application to the model with two independent fractional Brownian motions

The purpose of the article is twofold. Firstly, we review some recent results on the maximum likelihood estimation in the regression model of the form $X_t = θG(t) + B_t$, where $B$ is a Gaussian process, $G(t)$ is a known function, and $θ$ is an unknown drift parameter. The estimation techniques for the cases of discrete-time and continuous-time observations are presented. As examples, models with fractional Brownian motion, mixed fractional Brownian motion, and sub-fractional Brownian motion are considered. Secondly, we study in detail the model with two independent fractional Brownian motions and apply the general results mentioned above to this model.

math.PR↗
arXiv2018

Existence and uniqueness of mild solution to stochastic heat equation with white and fractional noises

We prove the existence and uniqueness of a mild solution for a class of non-autonomous parabolic mixed stochastic partial differential equations defined on a bounded open subset $D \subset \mathbb{R}^d$ and involving standard and fractional $L^2(D)$-valued Brownian motions. We assume that the coefficients are homogeneous, Lipschitz continuous and the coefficient at the fractional Brownian motion is an affine function.

math.PR↗
arXiv2017

Stochastic representation and pathwise properties of fractional Cox-Ingersoll-Ross process

We consider the fractional Cox-Ingersoll-Ross process satisfying the stochastic differential equation (SDE) $dX_t = aX_t\,dt + σ\sqrt{X_t}\,dB^H_t$ driven by a fractional Brownian motion (fBm) with Hurst parameter exceeding $\frac{2}{3}$. The integral $\int_0^t\sqrt{X_s}dB^H_s$ is considered as a pathwise integral and is equal to the limit of Riemann-Stieltjes integral sums. It is shown that the fractional Cox-Ingersoll-Ross process is a square of the fractional Ornstein-Uhlenbeck process until the first zero hitting. Based on that, we consider the square of the fractional Ornstein-Uhlenbeck process with an arbitrary Hurst index and prove that until its first zero hitting it satisfies the specified SDE if the integral $\int_0^t\sqrt{X_s}\,dB^H_s$ is defined as a pathwise Stratonovich integral. Therefore, the question about the first zero hitting time of the Cox-Ingersoll-Ross process, which matches the first zero hitting moment of the fractional Ornstein-Uhlenbeck process, is natural. Since the latter is a Gaussian process, it is proved by the estimates for distributions of Gaussian processes that for $a<0$ the probability of hitting zero in finite time is equal to 1, and in case of $a>0$ it is positive but less than 1. The upper bound for this probability is given.

math.PR↗
arXiv2017

Drift parameter estimation in stochastic differential equation with multiplicative stochastic volatility

We consider a stochastic differential equation of the form \[dX_t=θa(t,X_t)\,dt+σ_1(t,X_t)σ_2(t,Y_t)\,dW_t\] with multiplicative stochastic volatility, where $Y$ is some adapted stochastic process. We prove existence--uniqueness results for weak and strong solutions of this equation under various conditions on the process $Y$ and the coefficients $a$, $σ_1$, and $σ_2$. Also, we study the strong consistency of the maximum likelihood estimator for the unknown parameter $θ$. We suppose that $Y$ is in turn a solution of some diffusion SDE. Several examples of the main equation and of the process $Y$ are provided supplying the strong consistency.

math.PR↗
arXiv2016

Maximum likelihood drift estimation for Gaussian process with stationary increments

The paper deals with the regression model $X_t = θt + B_t$, $t\in[0, T ]$, where $B=\{B_t, t\geq 0\}$ is a centered Gaussian process with stationary increments. We study the estimation of the unknown parameter $θ$ and establish the formula for the likelihood function in terms of a solution to an integral equation. Then we find the maximum likelihood estimator and prove its strong consistency. The results obtained generalize the known results for fractional and mixed fractional Brownian motion.

math.PR↗
arXiv2016

Hypothesis testing of the drift parameter sign for fractional Ornstein-Uhlenbeck process

We consider the fractional Ornstein-Uhlenbeck process with an unknown drift parameter and known Hurst parameter $H$. We propose a new method to test the hypothesis of the sign of the parameter and prove the consistency of the test. Contrary to the previous works, our approach is applicable for all $H\in(0,1)$. We also study the estimators for drift parameter for continuous and discrete observations and prove their strong consistency for all $H\in(0,1)$.

math.PR↗
arXiv2016

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.

math.PR↗
arXiv2015

A generalisation of the fractional Brownian field based on non-Euclidean norms

We explore a generalisation of the Lévy fractional Brownian field on the Euclidean space based on replacing the Euclidean norm with another norm. A characterisation result for admissible norms yields a complete description of all self-similar Gaussian random fields with stationary increments. Several integral representations of the introduced random fields are derived. In a similar vein, several non-Euclidean variants of the fractional Poisson field are introduced and it is shown that they share the covariance structure with the fractional Brownian field and converge to it. The shape parameters of the Poisson and Brownian variants are related by convex geometry transforms, namely the radial $p$th mean body and the polar projection transforms.

math.PR↗
arXiv2015

Consistency of the drift parameter estimator for the discretized fractional Ornstein-Uhlenbeck process with Hurst index $H\in(0,\frac12)$

We consider Langevin equation involving fractional Brownian motion with Hurst index $H\in(0,\frac12)$. Its solution is the fractional Ornstein-Uhlenbeck process and with unknown drift parameter $θ$. We construct the estimator that is similar in form to maximum likelihood estimator for Langevin equation with standard Brownian motion. Observations are discrete in time. It is assumed that the interval between observations is $n^{-1}$, i.e. tends to zero (high frequency data) and the number of observations increases to infinity as $n^m$ with $m>1$. It is proved that for positive $θ$ the estimator is strongly consistent for any $m>1$ and for negative $θ$ it is consistent when $m>\frac{1}{2H}$.

math.PR↗