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Yuliya Mishura

Publications and source records attributed to Yuliya Mishura.

At least 37 records · Page 2Linked to original sources
arXiv2024

Properties of the entropic risk measure EVaR in relation to selected distributions

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, Gamma, Laplace, exponential, chi-squared, inverse Gaussian distribution and normal inverse Gaussian distribution with the help of Lambert function that is a special function, generally speaking, with two branches.

q-fin.RM↗
arXiv2024

Properties of Shannon and Rényi entropies of the Poisson distribution as the functions of intensity parameter

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 parameter $α>0$, we can characterize it as nontrivial. The proof is based on application of Karamata's inequality to the terms of Poisson distribution.

cs.IT↗
arXiv2023

A class of infinite-dimensional Gaussian processes defined through generalized fractional operators

The generalization of fractional Brownian motion in infinite-dimensional white and grey noise spaces has been recently carried over, following the Mandelbrot-Van Ness representation, through Riemann-Liouville type fractional operators. Our aim is to extend this construction by means of general fractional derivatives and integrals, which we define through Bernstein functions. According to the conditions satisfied by the latter, some properties of these processes (such as continuity, local times, variance asymptotics and persistence) are derived. On the other hand, they are proved to display short- or long-range dependence, if obtained by means of a derivative or an integral, respectively, regardless of the Bernstein function chosen. Moreover, this kind of construction allows us to define the corresponding noise and to derive an Ornstein-Uhlenbeck type process, as solution of an integral equation.

math.PR↗
arXiv2023

Parameter estimation in rough Bessel model

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.

math.PR↗
arXiv2023

Low-dimensional Cox-Ingersoll-Ross process

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 motion to derive this equation, which contains a term characterized by the local time of the corresponding reflected Brownian motion. Additionally, we establish a new connection between low-dimensional CIR processes and reflected Ornstein-Uhlenbeck (ROU) processes, providing a new representation of Skorokhod reflection functions.

math.PR↗
arXiv2023

Gaussian Volterra processes: asymptotic growth and statistical estimation

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 Volterra process under consideration. We construct a strongly consistent estimator and investigate its asymptotic properties. Namely, we prove that it has the Cauchy asymptotic distribution.

math.ST↗
arXiv2023

Entropy and alternative entropy functionals of fractional Gaussian noise as the functions of Hurst index

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 analytically at high dimensions, we also consider simple alternative entropy functionals, whose behavior, on the one hand, mimics the behavior of entropy and, on the other hand, is not difficult to study. Asymptotic behavior of the normalized entropy (so called entropy rate) is also studied for the entropy and for the alternative functionals.

math.PR↗
arXiv2022

Asymptotic expansion of an estimator for the Hurst coefficient

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.

math.ST↗
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↗
arXiv2022

Drift-implicit Euler scheme for sandwiched processes driven by Hölder noises

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, $λ\in(0,1)$. We prove that, under some mild moment assumptions on the Hölder constant of the noise, the $L^r(Ω;L^\infty([0,T]))$-rate of convergence is equal to $λ$. To exemplify, we consider numerical schemes for the generalized Cox--Ingersoll-Ross and Tsallis--Stariolo--Borland models. The results are illustrated by simulations.

math.PR↗
arXiv2021

Sandwiched SDEs with unbounded drift driven by Hölder noises

We study a stochastic differential equation with an unbounded drift and general Hölder continuous noise of an arbitrary order. 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 additional assumptions on the noise, we prove that the solution has moments of all orders. We complete the study providing a numerical scheme for the solution. As an illustration of our results and motivation for applications, we suggest two stochastic volatility models which we regard as generalizations of the CIR and CEV processes.

math.PR↗
arXiv2021

Standard and fractional reflected Ornstein-Uhlenbeck processes as the limits of square roots of Cox-Ingersoll-Ross processes

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 mean reversion parameter tends to either $σ^2/4$ (in the standard case) or to $0$ (in the fractional case). This also allows to obtain a new representation of the reflection function of the ROU as the limit of integral functionals of the CIR processes. The results of the paper are illustrated by simulations.

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↗
arXiv2021

Convexity and robustness of the Rényi entropy

We study convexity properties of Rényi entropy as function of $α>0$ 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 the number of events increases to $\infty$ and prove that limit of Rényi entropy of binomial distribution is equal to Rényi entropy of Poisson distribution.

math.PR↗
arXiv2021

Divergence of an integral of a process with small ball estimate

The paper contains sufficient conditions on the function $f$ and the stochastic process $X$ that supply the rate of divergence of the integral functional $\int_0^Tf(X_t)^2dt$ at the rate $T^{1-ε}$ as $T\to\infty$ for every $ε>0$. These conditions include so called small ball estimates which are discussed in detail. Statistical applications are provided.

math.PR↗
arXiv2020

Convergence results for the Time-Changed fractional Ornstein-Uhlenbeck processes

In this paper we study some convergence results concerning the one-dimensional distribution of a time-changed fractional Ornstein-Uhlenbeck process. In particular, we establish that, despite the time change, the process admits a Gaussian limit random variable. On the other hand, we prove that the process converges towards the time-changed Ornstein-Uhlenbeck as the Hurst index $H \to 1/2^+$, with locally uniform convergence of one-dimensional distributions. Moreover, we also achieve convergence in the Skorohod $J_1$-topology of the time-changed fractional Ornstein-Uhlenbeck process as $H \to 1/2^+$ in the space of càdlàg functions. Finally, we exploit some convergence properties of mild solutions of a generalized Fokker-Planck equation associated to the aforementioned processes, as $H \to 1/2^+$.

math.PR↗
arXiv2020

The Fokker-Planck equation for the time-changed fractional Ornstein-Uhlenbeck process

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 prove the weak maximum principle for strong solutions of the aforementioned equation and then a uniqueness result.

math.PR↗