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Description of the symmetric convex random closed sets as zonotopes from their Feret diameters

Saïd Rahmani Jean-Charles Pinoli Johan Debayle

https://doi.org/10.15559/16-VMSTA70

Pub. online: 3 Jan 2017 Type: Research Article

Open Access

Journal: Modern Stochastics: Theory and Applications Volume 3, Issue 4 (2016), pp. 325–364

Abstract

In this paper, the 2-D random closed sets (RACS) are studied by means of the Feret diameter, also known as the caliper diameter. More specifically, it is shown that a 2-D symmetric convex RACS can be approximated as precisely as we want by some random zonotopes (polytopes formed by the Minkowski sum of line segments) in terms of the Hausdorff distance. Such an approximation is fully defined from the Feret diameter of the 2-D convex RACS. Particularly, the moments of the random vector representing the face lengths of the zonotope approximation are related to the moments of the Feret diameter random process of the RACS.

An estimate for an expectation of the simultaneous renewal for time-inhomogeneous Markov chains

Vitaliy Golomoziy

https://doi.org/10.15559/16-VMSTA68

Pub. online: 23 Dec 2016 Type: Research Article

Open Access

Journal: Modern Stochastics: Theory and Applications Volume 3, Issue 4 (2016), pp. 315–323

Abstract

In this paper, we consider two time-inhomogeneous Markov chains ${X_{t}^{(l)}}$, $l\in \{1,2\}$, with discrete time on a general state space. We assume the existence of some renewal set C and investigate the time of simultaneous renewal, that is, the first positive time when the chains hit the set C simultaneously. The initial distributions for both chains may be arbitrary. Under the condition of stochastic domination and nonlattice condition for both renewal processes, we derive an upper bound for the expectation of the simultaneous renewal time. Such a bound was calculated for two time-inhomogeneous birth–death Markov chains.

Approximation of solutions of SDEs driven by a fractional Brownian motion, under pathwise uniqueness

Oussama El Barrimi Youssef Ouknine

https://doi.org/10.15559/16-VMSTA69

Pub. online: 20 Dec 2016 Type: Research Article

Open Access

Journal: Modern Stochastics: Theory and Applications Volume 3, Issue 4 (2016), pp. 303–313

Abstract

Our aim in this paper is to establish some strong stability properties of a solution of a stochastic differential equation driven by a fractional Brownian motion for which the pathwise uniqueness holds. The results are obtained using Skorokhod’s selection theorem.

Goodness-of-fit test in a multivariate errors-in-variables model

A X = B

Alexander Kukush Yaroslav Tsaregorodtsev

https://doi.org/10.15559/16-VMSTA67

Pub. online: 20 Dec 2016 Type: Research Article

Open Access

Journal: Modern Stochastics: Theory and Applications Volume 3, Issue 4 (2016), pp. 287–302

Abstract

We consider a multivariable functional errors-in-variables model $AX\approx B$, where the data matrices A and B are observed with errors, and a matrix parameter X is to be estimated. A goodness-of-fit test is constructed based on the total least squares estimator. The proposed test is asymptotically chi-squared under null hypothesis. The power of the test under local alternatives is discussed.

Drift parameter estimation in stochastic differential equation with multiplicative stochastic volatility

Meriem Bel Hadj Khlifa Yuliya Mishura Kostiantyn Ralchenko Mounir Zili

https://doi.org/10.15559/16-VMSTA66

Pub. online: 13 Dec 2016 Type: Research Article

Open Access

Journal: Modern Stochastics: Theory and Applications Volume 3, Issue 4 (2016), pp. 269–285

Abstract

We consider a stochastic differential equation of the form

\[ dX_{t}=\theta a(t,X_{t})\hspace{0.1667em}dt+\sigma _{1}(t,X_{t})\sigma _{2}(t,Y_{t})\hspace{0.1667em}dW_{t}\]

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, $\sigma _{1}$, and $\sigma _{2}$. 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.

Workshop “Fractality and Fractionality”

Yuliya Mishura Georgiy Shevchenko

https://doi.org/10.15559/16-VMSTA65

Pub. online: 15 Nov 2016 Type: Meeting Report

Open Access

Journal: Modern Stochastics: Theory and Applications Volume 3, Issue 3 (2016), pp. 209–211

Abstract

This short note is devoted to the “Fractality and Fractionality” workshop, held on 17–20 May 2016 in Lorentz Center (Leiden, Netherlands).

Averaged deviations of Orlicz processes and majorizing measures

Rostyslav Yamnenko

https://doi.org/10.15559/16-VMSTA64

Pub. online: 11 Nov 2016 Type: Research Article

Open Access

Journal: Modern Stochastics: Theory and Applications Volume 3, Issue 3 (2016), pp. 249–268

Abstract

This paper is devoted to investigation of supremum of averaged deviations $|X(t)-f(t)-\int _{\mathbb{T}}(X(u)-f(u))\hspace{0.1667em}\mathrm{d}\mu (u)/\mu (\mathbb{T})|$ of a stochastic process from Orlicz space of random variables using the method of majorizing measures. An estimate of distribution of supremum of deviations $|X(t)-f(t)|$ is derived. A special case of the $L_{q}$ space is considered. As an example, the obtained results are applied to stochastic processes from the $L_{2}$ space with known covariance functions.

Stochastic wave equation in a plane driven by spatial stable noise

Larysa Pryhara Georgiy Shevchenko

https://doi.org/10.15559/16-VMSTA62

Pub. online: 8 Nov 2016 Type: Research Article

Open Access

Journal: Modern Stochastics: Theory and Applications Volume 3, Issue 3 (2016), pp. 237–248

Abstract

The main object of this paper is the planar wave equation

\[ \bigg(\frac{{\partial }^{2}}{\partial {t}^{2}}-{a}^{2}\varDelta \bigg)U(x,t)=f(x,t),\hspace{1em}t\ge 0,\hspace{2.5pt}x\in {\mathbb{R}}^{2},\]

with random source f. The latter is, in certain sense, a symmetric α-stable spatial white noise multiplied by some regular function σ. We define a candidate solution U to the equation via Poisson’s formula and prove that the corresponding expression is well defined at each point almost surely, although the exceptional set may depend on the particular point $(x,t)$. We further show that U is Hölder continuous in time but with probability 1 is unbounded in any neighborhood of each point where σ does not vanish. Finally, we prove that U is a generalized solution to the equation.

A limit theorem for singular stochastic differential equations

Andrey Pilipenko Yuriy Prykhodko

https://doi.org/10.15559/16-VMSTA63

Pub. online: 8 Nov 2016 Type: Research Article

Open Access

Journal: Modern Stochastics: Theory and Applications Volume 3, Issue 3 (2016), pp. 223–235

Abstract

We study the weak limits of solutions to SDEs

\[ dX_{n}(t)=a_{n}\big(X_{n}(t)\big)\hspace{0.1667em}dt+dW(t),\]

where the sequence $\{a_{n}\}$ converges in some sense to $(c_{-}\mathbb{1}_{x<0}+c_{+}\mathbb{1}_{x>0})/x+\gamma \delta _{0}$. Here $\delta _{0}$ is the Dirac delta function concentrated at zero. A limit of $\{X_{n}\}$ may be a Bessel process, a skew Bessel process, or a mixture of Bessel processes.

On spectra of probability measures generated by GLS-expansions

Marina Lupain

https://doi.org/10.15559/16-VMSTA61

Pub. online: 26 Oct 2016 Type: Research Article

Open Access

Journal: Modern Stochastics: Theory and Applications Volume 3, Issue 3 (2016), pp. 213–221

Abstract

We study properties of distributions of random variables with independent identically distributed symbols of generalized Lüroth series (GLS) expansions (the family of GLS-expansions contains Lüroth expansion and $Q_{\infty }$- and ${G_{\infty }^{2}}$-expansions). To this end, we explore fractal properties of the family of Cantor-like sets $C[\mathit{GLS},V]$ consisting of real numbers whose GLS-expansions contain only symbols from some countable set $V\subset N\cup \{0\}$, and derive exact formulae for the determination of the Hausdorff–Besicovitch dimension of $C[\mathit{GLS},V]$. Based on these results, we get general formulae for the Hausdorff–Besicovitch dimension of the spectra of random variables with independent identically distributed GLS-symbols for the case where all but countably many points from the unit interval belong to the basis cylinders of GLS-expansions.

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