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Convergence rate of CLT for the drift estimation of sub-fractional Ornstein–Uhlenbeck process of second kind

Maoudo Faramba Baldé Khalifa Es-Sebaiy

https://doi.org/10.15559/21-VMSTA179

Pub. online: 20 May 2021 Type: Research Article

Open Access

Journal: Modern Stochastics: Theory and Applications Volume 8, Issue 3 (2021), pp. 329–347

Abstract

In this paper, we deal with an Ornstein–Uhlenbeck process driven by sub-fractional Brownian motion of the second kind with Hurst index $H\in (\frac{1}{2},1)$. We provide a least squares estimator (LSE) of the drift parameter based on continuous-time observations. The strong consistency and the upper bound $O(1/\sqrt{n})$ in Kolmogorov distance for central limit theorem of the LSE are obtained. We use a Malliavin–Stein approach for normal approximations.

On nonnegative solutions of SDDEs with an application to CARMA processes

Mikkel Slot Nielsen

Victor Rohde

https://doi.org/10.15559/21-VMSTA177

Pub. online: 31 Mar 2021 Type: Research Article

Open Access

Journal: Modern Stochastics: Theory and Applications Volume 8, Issue 3 (2021), pp. 309–328

Abstract

This note provides a simple sufficient condition ensuring that solutions of stochastic delay differential equations (SDDEs) driven by subordinators are nonnegative. While, to the best of our knowledge, no simple nonnegativity conditions are available in the context of SDDEs, we compare our result to the literature within the subclass of invertible continuous-time ARMA (CARMA) processes. In particular, we analyze why our condition cannot be necessary for CARMA($p,q$) processes when $p=2$, and we show that there are various situations where our condition applies while existing results do not as soon as $p\ge 3$. Finally, we extend the result to a multidimensional setting.

Statistical inference for nonergodic weighted fractional Vasicek models

Khalifa Es-Sebaiy

Mishari Al-Foraih Fares Alazemi

https://doi.org/10.15559/21-VMSTA176

Pub. online: 26 Mar 2021 Type: Research Article

Open Access

Journal: Modern Stochastics: Theory and Applications Volume 8, Issue 3 (2021), pp. 291–307

Abstract

A problem of drift parameter estimation is studied for a nonergodic weighted fractional Vasicek model defined as $d{X_{t}}=\theta (\mu +{X_{t}})dt+d{B_{t}^{a,b}}$, $t\ge 0$, with unknown parameters $\theta >0$, $\mu \in \mathbb{R}$ and $\alpha :=\theta \mu $, whereas ${B^{a,b}}:=\{{B_{t}^{a,b}},t\ge 0\}$ is a weighted fractional Brownian motion with parameters $a>-1$, $|b|<1$, $|b|<a+1$. Least square-type estimators $({\widetilde{\theta }_{T}},{\widetilde{\mu }_{T}})$ and $({\widetilde{\theta }_{T}},{\widetilde{\alpha }_{T}})$ are provided, respectively, for $(\theta ,\mu )$ and $(\theta ,\alpha )$ based on a continuous-time observation of $\{{X_{t}},\hspace{2.5pt}t\in [0,T]\}$ as $T\to \infty $. The strong consistency and the joint asymptotic distribution of $({\widetilde{\theta }_{T}},{\widetilde{\mu }_{T}})$ and $({\widetilde{\theta }_{T}},{\widetilde{\alpha }_{T}})$ are studied. Moreover, it is obtained that the limit distribution of ${\widetilde{\theta }_{T}}$ is a Cauchy-type distribution, and ${\widetilde{\mu }_{T}}$ and ${\widetilde{\alpha }_{T}}$ are asymptotically normal.

Chaos expansion of uniformly distributed random variables and application to number theory

Ciprian Tudor

https://doi.org/10.15559/21-VMSTA172

Pub. online: 26 Mar 2021 Type: Research Article

Open Access

Journal: Modern Stochastics: Theory and Applications Volume 8, Issue 2 (2021), pp. 275–289

Abstract

The chaos expansion of a random variable with uniform distribution is given. This decomposition is applied to analyze the behavior of each chaos component of the random variable $\log \zeta $ on the so-called critical line, where ζ is the Riemann zeta function. This analysis gives a better understanding of a famous theorem by Selberg.

Existence and uniqueness of weak solution to a three-dimensional stochastic modified-Leray-alpha model of fluid turbulence

Ridha Selmi Rim Nasfi

https://doi.org/10.15559/21-VMSTA175

Pub. online: 16 Mar 2021 Type: Research Article

Open Access

Journal: Modern Stochastics: Theory and Applications Volume 8, Issue 1 (2021), pp. 115–137

Abstract

In this paper, we study the stochastic three-dimensional modified Leray-alpha model arising from the turbulent flows of fluids. We prove the existence of the probabilistic weak solution under the non-Lipschitz condition for the nonlinear forcing terms. We also discuss its uniqueness.

Note on local mixing techniques for stochastic differential equations

Alexander Veretennikov

https://doi.org/10.15559/21-VMSTA174

Pub. online: 16 Mar 2021 Type: Research Article

Open Access

Journal: Modern Stochastics: Theory and Applications Volume 8, Issue 1 (2021), pp. 1–15

Abstract

The paper discusses several techniques which may be used for applying the coupling method to solutions of stochastic differential equations (SDEs). The coupling techniques traditionally consist of two components: one is local mixing, the other is recurrence. Often in the articles they do not split. Yet, they are quite different in their nature, and this paper separates them, concentrating only on the former.

Most of the techniques discussed here work in dimension $d\ge 1$, although, in $d=1$ there is one additional option to use intersections of trajectories, which requires nothing but the strong Markov property and nondegeneracy of the diffusion coefficient. In dimensions $d>1$ it is possible to use embedded Markov chains either by considering discrete times $n=0,1,\dots $, or by arranging special stopping time sequences and to use the local Markov–Dobrushin (MD) condition, which is one of the most efficient versions of local mixing. Further applications may be based on one or another version of the MD condition; respectively, this paper is devoted to various methods of verifying one or another form of it.

Long-time behavior of a nonautonomous stochastic predator–prey model with jumps

Olga Borysenko Oleksandr Borysenko

https://doi.org/10.15559/21-VMSTA173

Pub. online: 8 Mar 2021 Type: Research Article

Open Access

Journal: Modern Stochastics: Theory and Applications Volume 8, Issue 1 (2021), pp. 17–39

Abstract

The existence and uniqueness of a global positive solution is proven for the system of stochastic differential equations describing a nonautonomous stochastic predator–prey model with a modified version of the Leslie–Gower term and Holling-type II functional response disturbed by white noise, centered and noncentered Poisson noises. Sufficient conditions are obtained for stochastic ultimate boundedness, stochastic permanence, nonpersistence in the mean, weak persistence in the mean and extinction of a solution to the considered system.

Investigation of sample paths properties for some classes of φ-sub-Gaussian stochastic processes

Olha Hopkalo Lyudmyla Sakhno

https://doi.org/10.15559/21-VMSTA171

Pub. online: 26 Jan 2021 Type: Research Article

Open Access

Journal: Modern Stochastics: Theory and Applications Volume 8, Issue 1 (2021), pp. 41–62

Abstract

This paper investigates sample paths properties of φ-sub-Gaussian processes by means of entropy methods. Basing on a particular entropy integral, we treat the questions on continuity and the rate of growth of sample paths. The obtained results are then used to investigate the sample paths properties for a particular class of φ-sub-Gaussian processes related to the random heat equation. We derive the estimates for the distribution of suprema of such processes and evaluate their rate of growth.

2010 Mathematics Subject Classification index

https://doi.org/10.15559/20-VMSTA74MI

Pub. online: 23 Dec 2020 Type: 2010 Mathematics Subject Classification Index

Open Access

Journal: Modern Stochastics: Theory and Applications Volume 7, Issue 4 (2020), pp. 473–476

Keywords index

https://doi.org/10.15559/20-VMSTA74KI

Pub. online: 23 Dec 2020 Type: Keywords Index

Open Access

Journal: Modern Stochastics: Theory and Applications Volume 7, Issue 4 (2020), pp. 471–472

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Online ISSN: 2351-6054
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