Yuliia Mishura

Yuliia Mishura
Taras Shevchenko National University of Kyiv | Київський національний університет імені Тараса Шевченка · Faculty of Mechanics and Mathematics

Doctor of Sciences

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500
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Publications

Publications (500)
Preprint
The paper is devoted to the properties of the entropy of the exponent-Wiener-integral fractional Gaussian process (EWIFG-process), that is a Wiener integral of the exponent with respect to fractional Brownian motion. Unlike fractional Brownian motion, whose entropy has very simple monotonicity properties in Hurst index, the behavior of the entropy...
Preprint
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We study the asymptotic behaviour of a properly normalized time-changed multidimensional Wiener process; the time change is given by an additive functional of the Wiener process itself. At the level of generators, the time change means that we consider the Laplace operator -- which generates a multidimensional Wiener process -- and multiply it by a...
Preprint
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We consider a continuous-time financial market with an asset whose price is modeled by a linear stochastic differential equation with drift and volatility switching driven by a uniformly ergodic jump Markov process with a countable state space (in fact, this is a Black-Scholes model with Markov switching). We construct a multiplicative scheme of se...
Article
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This paper studies two related stochastic processes driven by Brownian motion: the Cox–Ingersoll–Ross (CIR) process and the Bessel process. We investigate their shared and distinct properties, focusing on time-asymptotic growth rates, distance between the processes in integral norms, and parameter estimation. The squared Bessel process is shown to...
Article
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The paper focuses on the Vasicek model driven by a tempered fractional Brownian motion. We derive the asymptotic distributions of the least-squares estimators (based on continuous-time observations) for the unknown drift parameters. This work continues the investigation by Mishura and Ralchenko (Fractal and Fractional, 8(2:79), 2024), where these e...
Preprint
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The paper extends the analysis of the entropies of the Poisson distribution with parameter $\lambda$. It demonstrates that the Tsallis and Sharma-Mittal entropies exhibit monotonic behavior with respect to $\lambda$, whereas two generalized forms of the R\'enyi entropy may exhibit "anomalous" (non-monotonic) behavior. Additionally, we examine the a...
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We calculate and analyze various entropy measures and their properties for selected probability distributions. The entropies considered include Shannon, R\'enyi, generalized R\'enyi, Tsallis, Sharma-Mittal, and modified Shannon entropy, along with the Kullback-Leibler divergence. These measures are examined for several distributions, including gamm...
Preprint
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This paper studies two related stochastic processes driven by Brownian motion: the Cox-Ingersoll-Ross (CIR) process and the Bessel process. We investigate their shared and distinct properties, focusing on time-asymptotic growth rates, distance between the processes in integral norms, and parameter estimation. The squared Bessel process is shown to...
Article
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The article is devoted to the life and scientific path of academician of the National Academy of Sciences of Ukraine, Нonored professor of Taras Shevchenko Kyiv National University, Mykola Oleksiyovych Perestyuk. In particular, the material contains a complete list of Mykola Oleksiyovych's students and the topics of their dissertations, a list of m...
Preprint
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We examine the one-sided and two-sided (bilateral) projections of an element of fractional Gaussian noise onto its neighboring elements. We establish several analytical results and conduct a numerical study to analyze the behavior of the coefficients of these projections as functions of the Hurst index and the number of neighboring elements used fo...
Article
We consider two types of entropy, namely, Shannon and Rényi entropies of the Poisson distribution, and establish their properties as the functions of intensity parameter. More precisely, we prove that both entropies increase with intensity. While for Shannon entropy the proof is comparatively simple, for Rényi entropy, which depends on additional p...
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In this short note, we introduce probabilistic Cauchy functional equations, specifically, functional equations of the following form: $$ f(X_1 + X_2) \stackrel{d}{=} f(X_1) + f(X_2), $$ where $X_1$ and $X_2$ represent two independent identically distributed real-valued random variables governed by a distribution $\mu$ having appropriate support on...
Preprint
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The paper focuses on the Vasicek model driven by a tempered fractional Brownian motion. We derive the asymptotic distributions of the least-squares estimators (based on continuous-time observations) for the unknown drift parameters. This work continues the investigation by Mishura and Ralchenko (Fractal and Fractional, 8(2:79), 2024), where these e...
Article
Full-text available
Entropic Value-at-Risk (EVaR) measure is a convenient coherent risk measure. Due to certain difficulties in finding its analytical representation, it was previously calculated explicitly only for the normal distribution. We succeeded to overcome these difficulties and to calculate Entropic Value-at-Risk (EVaR) measure for Poisson, compound Poisson,...
Chapter
We introduce the spaces of test functions and generalized functions. Du Bois—Reymond lemma is formulated. Differential operators on the spaces of generalized functions are treated. Variety of generalized functions are considered to illustrate theoretical part.
Chapter
Fredholm and Volterra integral equations are considered. Fredholm alternative is formulated. Problems are devoted to solving of integral equations of different types.
Chapter
The notions of regular point, resolvent set, spectrum, spectral radius, eigenvalue and eigenfunction are introduced. The components of spectrum are considered. Spectra of different operators are investigated.
Chapter
We study different types (uniform, strong, weak) of convergence of linear continuous operators. The problems are devoted to the examples of convergent (non-convergent) series of operators.
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We consider the notions of linear space, norm, normed linear space (NLS), Banach space. Various examples of Banach spaces are treated. The problems help to understand what functions are (are not) the norms on respective spaces.
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Hermitian, adjoint, self-adjoint, normal, nonnegative, unitary, and other types of operators in Hilbert spaces are introduced and illustrated by the variety of examples.
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Precompact and compact sets in the metric spaces are introduced, together with the notion of compact operator. Hausdorff criterion and Ascoli-Arzelà theorem are formulated. Problems explain how to establish or controvert the compactness.
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We consider the notions of linear, continuous and bounded operator. For linear bounded operator the norm is introduced. The problems are devoted to the examples of operators and their basic properties.
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We consider the notions of linear, continuous and bounded functional. Numerous examples of linear continuous functionals are provided and their norms are calculated.
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The properties of spectrum of compact operator in infinite-dimensional Banach space are established. Hilbert-Schmidt theorem is formulated. Solving the problems, one can study the structure of spectrum of various compact operators.
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Banach-Steinhaus theorem is formulated and different notions of convergence of linear continuous functionals and elements in Banach spaces are studied.
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Let H be a complex linear space. A function \((\cdot ,\cdot ):H\times H\to \mathbb {C}\) is called a scalar product (or an inner product).
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The notion of extension in continuity of a linear continuous functional is introduced an famous Hahn-Banach theorem is formulated, together with several corollaries. Problems illustrate how to construct the extensions.
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Notions of algebraic inverse operator and continuously invertible operator are studied. Criterion of continuous invertibility is formulated. Problems are devoted to the examples of different types of invertibility and the properties of inverse operators.
Article
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The Gaussian-Volterra process with a linear kernel is considered, its properties are established and projection coefficients are explicitly calculated, i.e. one of possible prediction problems related to Gaussian processes is solved.
Article
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Tempered fractional Brownian motion (TFBM) and tempered fractional Brownian motion of the second kind (TFBMII) modify the power-law kernel in the moving average representation of fractional Brownian motion by introducing exponential tempering. We construct least-square estimators for the unknown drift parameters within Vasicek models that are drive...
Article
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The present paper investigates Cox-Ingersoll-Ross (CIR) processes of dimension less than 1, with a focus on obtaining an equation of a new type including local times for the square root of the CIR process. To derive this equation, we utilize the fact that non-negative diffusion processes can be obtained by the transformation of time and scale of a...
Article
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This paper is devoted to the analysis of court cases based on multiple sentences that represent plaintiff's claim, claim motivation and defendant's response. Based on these parameters we classify a given case into one of seven categories designed for our task and then predict its decision in the first court's instance. We use fine-tuned XLM\RoBERTa...
Article
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On September 28, 2023, Mykhailo Moklyachuk, Doctor of Physical and Mathematical Sciences, Professor, Laureate of the State Prize of Ukraine in Education, Honored Worker of Science and Technology of Ukraine, and Academician of the Academy of Sciences of the Higher School of Ukraine, celebrated his 75th birthday. His scientific research is devoted to...
Article
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We consider five types of entropies for Gaussian distribution: Shannon, Rényi, generalized Rényi, Tsallis and Sharma–Mittal entropy, establishing their interrelations and their properties as the functions of parameters. Then, we consider fractional Gaussian processes, namely fractional, subfractional, bifractional, multifractional and tempered frac...
Article
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In this paper, we present a comprehensive survey of continuous stochastic volatility models, discussing their historical development and the key stylized facts that have driven the field. Special attention is dedicated to fractional and rough methods: without advocating for either roughness or long memory, we outline the motivation behind them and...
Article
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Asymptotic expansion is presented for an estimator of the Hurst coefficient of a fractional Brownian motion. We first derive the expansion formula of the principal term of the error of the estimator using a recently developed theory of asymptotic expansion of the distribution of Wiener functionals, and utilize the perturbation method on the obtaine...
Preprint
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In this paper, we present a comprehensive survey of continuous stochastic volatility models, discussing their historical development and the key stylized facts that have driven the field. Special attention is dedicated to fractional and rough methods: without advocating for either roughness or long memory, we outline the motivation behind them and...
Article
Full-text available
In this paper, we construct consistent statistical estimators of the Hurst index, volatility coefficient, and drift parameter for Bessel processes driven by fractional Brownian motion with H<1/2. As an auxiliary result, we also prove the continuity of the fractional Bessel process. The results are illustrated with simulations.
Preprint
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In this paper, we construct consistent statistical estimators of the Hurst index, volatility coefficient, and drift parameter for Bessel processes driven by fractional Brownian motion with $H<1/2$. As an auxiliary result, we also prove the continuity of the fractional Bessel process. The results are illustrated with simulations.
Article
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This paper is devoted to the study of the properties of entropy as a function of the Hurst index, which corresponds to the fractional Gaussian noise. Since the entropy of the Gaussian vector depends on the determinant of the covariance matrix, and the behavior of this determinant as a function of the Hurst index is rather difficult to study analyti...
Article
The paper is devoted to three-parametric self-similar Gaussian Volterra processes that generalize fractional Brownian motion. We study the asymptotic growth of such processes and the properties of long- and short-range dependence. Then we consider the problem of the drift parameter estimation for Ornstein–Uhlenbeck process driven by Gaussian Volter...
Preprint
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We introduce fractional diffusion Bessel process with Hurst index $H\in(0,\frac12)$, derive a stochastic differential equation for it, and study the asymptotic properties of its sample paths.
Article
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The article is devoted to the drift parameters estimation in the Cox–Ingersoll–Ross model. We obtain the rate of convergence in probability of the maximum likelihood estimators based on the continuous-time estimators. Then we introduce the discrete versions of these estimators and investigate their asymptotic behavior. In particular, we establish t...
Preprint
Full-text available
The present paper investigates Cox-Ingersoll-Ross (CIR) processes of dimension less than 1, with a focus on obtaining an equation of a new type including local times for the square root of the CIR process. We utilize the fact that non-negative diffusion processes can be obtained by the transformation of time and scale of some reflected Brownian mot...
Article
We study a stochastic differential equation with an unbounded drift and general Hölder continuous noise of order $\lambda \in (0,1)$ . The corresponding equation turns out to have a unique solution that, depending on a particular shape of the drift, either stays above some continuous function or has continuous upper and lower bounds. Under some mil...
Preprint
Full-text available
This paper is devoted to the study of the properties of entropy as a function of the Hurst index, which corresponds to the fractional Gaussian noise. Since the entropy of the Gaussian vector depends on the determinant of the covariance matrix, and the behavior of this determinant as a function of the Hurst index is rather difficult to study analyti...
Chapter
We study Volterra processes Xt=∫0tK(t,s)dWs, where W is a standard Wiener process, and the kernel has the form K(t,s)=a(s)∫stb(u)c(u-s)du. This form generalizes the Volterra kernel for fractional Brownian motion (fBm) with Hurst index H>1/2. We establish smoothness properties of X, including continuity and Hölder property. It happens that its Hölde...
Chapter
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...
Article
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In this paper, we consider the concept of invariant sets of inhomogeneous stochastic differential equations with jumps. For certain classes of systems of the second order of inhomogeneous stochastic differential equations with jumps the necessary and sufficient conditions for the invariance of the corresponding surfaces are established. The obtained re...
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Pages of the article in the issue: 11 - 21 Language of the article: Ukrainian
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In this paper, we analyze the drift-implicit (or backward) Euler numerical scheme for a class of stochastic differential equations with unbounded drift driven by an arbitrary λ-Hölder continuous process, λ ∈ (0,1). We prove that, under some mild moment assumptions on the Hölder constant of the noise, the Lr(Ω;L∞([0,T]))\documentclass[12pt]{minimal}...
Article
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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 thes...
Preprint
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We introduce a new model of financial market with stochastic volatility driven by an arbitrary H\"older continuous Gaussian Volterra process. The distinguishing feature of the model is the form of the volatility equation which ensures the solution to be ``sandwiched'' between two arbitrary H\"older continuous functions chosen in advance. We discuss...
Preprint
Asymptotic expansion is presented for an estimator of the Hurst coefficient of a fractional Brownian motion. For this, a recently developed theory of asymptotic expansion of the distribution of Wiener functionals is applied. The effects of the asymptotic expansion are demonstrated by numerical studies.
Preprint
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 thes...
Article
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The article deals with numerical estimation of the drift parameter in the continuous-time linear model with two independent fractional Brownian motions. The main focus is given to the computational difficulties of the maximum likelihood approach, in particular, to the construction of the approximate solution to the Fredholm integral equation of the...
Article
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In this paper the study of a three-parametric class of Gaussian Volterra processes is continued. This study was started in Part I of the present paper. The class under consideration is a generalization of a fractional Brownian motion that is in fact a one-parametric process depending on Hurst index H. On the one hand, the presence of three paramete...
Article
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The paper is devoted to the basic properties of fractional integrals. It is a survey of the well-known properties of fractional integrals, however, the authors tried to present the known information about fractional integrals as short and transparently as possible. We introduce fractional integrals on the compact interval and on the semi-axes, cons...
Preprint
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In this paper, we analyze the drift-implicit (or backward) Euler numerical scheme for a class of stochastic differential equations with unbounded drift driven by an arbitrary $\lambda$-H\"older continuous process, $\lambda\in(0,1)$. We prove that, under some mild moment assumptions on the H\"older constant of the noise, the $L^r(\Omega;L^\infty([0,...
Article
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We investigate the mixed fractional Brownian motion with trend of the form \(X_t = \theta t + \sigma W_t + \kappa B^H_t\), driven by a standard Brownian motion W and a fractional Brownian motion \(B^H\) with Hurst parameter H. We develop and compare two approaches to estimation of four unknown parameters \(\theta \), \(\sigma \), \(\kappa \) and H...
Article
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The stochastic process of the form \[ {X_{t}}={\int _{0}^{t}}{s^{\alpha }}\left({\int _{s}^{t}}{u^{\beta }}{(u-s)^{\gamma }}\hspace{0.1667em}du\right)\hspace{0.1667em}d{W_{s}}\] is considered, where W is a standard Wiener process, $\alpha >-\frac{1}{2}$, $\gamma >-1$, and $\alpha +\beta +\gamma >-\frac{3}{2}$. It is proved that the process X is...
Article
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In this paper, we establish a new connection between Cox–Ingersoll–Ross (CIR) and reflected Ornstein–Uhlenbeck (ROU) models driven by either a standard Wiener process or a fractional Brownian motion with H>12. We prove that, with probability 1, the square root of the CIR process converges uniformly on compacts to the ROU process as the mean reversi...
Article
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The present paper investigates the effects of tempering the power law kernel of the moving average representation of a fractional Brownian motion (fBm) on some local and global properties of this Gaussian stochastic process. Tempered fractional Brownian motion (TFBM) and tempered fractional Brownian motion of the second kind (TFBMII) are the proces...
Article
We consider a stochastic differential equation of the form drt=(a−brt)dt+σrtβdWt, where a, b and σ are positive constants, β∈(12,1). We study the estimation of an unknown drift parameter (a,b) by continuous observations of a sample path {rt,t∈[0,T]}. We prove the strong consistency and asymptotic normality of the maximum likelihood estimator. We pr...
Article
The paper contains sufficient conditions on the function f and the stochastic process X that supply the rate of divergence of the integral functional ∫0Tf(Xt)2dt at the rate T1−ε as T→∞ for every ε>0. These conditions include so called small ball estimates which are discussed in detail. Statistical applications are provided.
Article
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The paper is devoted to the rate of convergence of integral sums of two different types to fractional integrals. The first theorem proves the H¨older property of fractional integrals of functions from various integral spaces. Then we estimate the rate of convergence of the integral sums of two types corresponding to the H¨older functions, to the re...
Preprint
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In this paper, we establish a new connection between Cox-Ingersoll-Ross (CIR) and reflected Ornstein-Uhlenbeck (ROU) models driven by either a standard Wiener process or a fractional Brownian motion with $H>\frac{1}{2}$. We prove that, with probability 1, the square root of the CIR process converges uniformly on compacts to the ROU process as the m...
Article
In this paper, we study some properties of the generalized Fokker–Planck equation induced by the time-changed fractional Ornstein–Uhlenbeck process. First of all, we exploit some sufficient conditions to show that a mild solution of such equation is actually a classical solution. Then, we discuss an isolation result for mild solutions. Finally, we...
Article
The article is devoted to the approximate solutions of the Fredholm integral equations of the second kind with the weak singular kernel that can have additional singularity in the numerator. We describe two problems that lead to such equations. They are the problem of minimization of small deviations and the entropy minimization problem. Both of th...
Article
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We study convexity properties of the Rényi entropy as function of $\alpha >0$ on finite alphabets. We also describe robustness of the Rényi entropy on finite alphabets, and it turns out that the rate of respective convergence depends on initial alphabet. We establish convergence of the disturbed entropy when the initial distribution is uniform but...
Article
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We study boundary non-crossing probabilities $$\begin{aligned} P_{f,u} := \mathrm {P}\big (\forall t\in {\mathbb {T}}\ X_t + f(t)\le u(t)\big ) \end{aligned}$$for a continuous centered Gaussian process X indexed by some arbitrary compact separable metric space \({\mathbb {T}}\). We obtain both upper and lower bounds for \(P_{f,u}\). The bounds are...
Preprint
We consider a stochastic differential equation of the form $dr_t = (a - b r_t) dt + \sigma r_t^\beta dW_t$, where $a$, $b$ and $\sigma$ are positive constants, $\beta\in(\frac12,1)$. 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 a...
Article
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In the high-frequency limit, conditionally expected increments of fractional Brownian motion converge to a white noise, shedding their dependence on the path history and the forecasting horizon and making dynamic optimisation problems tractable. We find an explicit formula for locally mean–variance optimal strategies and their performance for an as...
Article
The paper is devoted to the existence of perpetual integral functionals ∫0∞f(X(t))dtfor several classes of d-dimensional of stochastic processes X(t). The method is very simple: we establish the conditions supplying that these functionals have a finite expectation. Examples of these classes include d-dimensional fractional Brownian motion having co...
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We study convexity properties of R\'{e}nyi entropy as function of $\alpha>0$ on finite alphabets. We also describe robustness of the R\'{e}nyi entropy on finite alphabets, and it turns out that the rate of respective convergence depends on initial alphabet. We establish convergence of the disturbed entropy when the initial distribution is uniform b...
Article
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We consider a fractional Ornstein-Uhlenbeck process involving a stochastic forcing term in the drift, as a solution of a linear stochastic differential equation driven by a fractional Brownian motion. For such process we specify mean and covariance functions, concentrating on their asymptotic behavior. This gives us a sort of short- or long-range d...