Help

Home
Search

Modern Stochastics: Theory and Applications

Submit your article Information Become a Peer-reviewer

Journal home
To appear
Current issue
All issues
More
Journal home To appear Current issue All issues

Detailed search

Title

Author

Types

Abstract

Keywords

Published

Pages

Volumes

Issues

DOI

Affiliation

Classifications

Search results 301

Order by:

Select: All None Download:

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.

Asymptotic behavior of homogeneous additive functionals of the solutions of Itô stochastic differential equations with nonregular dependence on parameter

Grigorij Kulinich Svitlana Kushnirenko Yuliia Mishura

https://doi.org/10.15559/16-VMSTA58

Pub. online: 4 Jul 2016 Type: Research Article

Open Access

Journal: Modern Stochastics: Theory and Applications Volume 3, Issue 2 (2016), pp. 191–208

Abstract

We study the asymptotic behavior of mixed functionals of the form $I_{T}(t)=F_{T}(\xi _{T}(t))+{\int _{0}^{t}}g_{T}(\xi _{T}(s))\hspace{0.1667em}d\xi _{T}(s)$, $t\ge 0$, as $T\to \infty $. Here $\xi _{T}(t)$ is a strong solution of the stochastic differential equation $d\xi _{T}(t)=a_{T}(\xi _{T}(t))\hspace{0.1667em}dt+dW_{T}(t)$, $T>0$ is a parameter, $a_{T}=a_{T}(x)$ are measurable functions such that $\left|a_{T}(x)\right|\le C_{T}$ for all $x\in \mathbb{R}$, $W_{T}(t)$ are standard Wiener processes, $F_{T}=F_{T}(x)$, $x\in \mathbb{R}$, are continuous functions, $g_{T}=g_{T}(x)$, $x\in \mathbb{R}$, are locally bounded functions, and everything is real-valued. The explicit form of the limiting processes for $I_{T}(t)$ is established under very nonregular dependence of $g_{T}$ and $a_{T}$ on the parameter T.

Simulation paradoxes related to a fractional Brownian motion with small Hurst index

Vitalii Makogin

https://doi.org/10.15559/16-VMSTA59

Pub. online: 4 Jul 2016 Type: Research Article

Open Access

Journal: Modern Stochastics: Theory and Applications Volume 3, Issue 2 (2016), pp. 181–190

Abstract

We consider the simulation of sample paths of a fractional Brownian motion with small values of the Hurst index and estimate the behavior of the expected maximum. We prove that, for each fixed N, the error of approximation $\mathbf{E}\max _{t\in [0,1]}{B}^{H}(t)-\mathbf{E}\max _{i=\overline{1,N}}{B}^{H}(i/N)$ grows rapidly to ∞ as the Hurst index tends to 0.

Randomly stopped sums with consistently varying distributions

Edita Kizinevič Jonas Sprindys Jonas Šiaulys

https://doi.org/10.15559/16-VMSTA60

Pub. online: 4 Jul 2016 Type: Research Article

Open Access

Journal: Modern Stochastics: Theory and Applications Volume 3, Issue 2 (2016), pp. 165–179

Abstract

Let $\{\xi _{1},\xi _{2},\dots \}$ be a sequence of independent random variables, and η be a counting random variable independent of this sequence. We consider conditions for $\{\xi _{1},\xi _{2},\dots \}$ and η under which the distribution function of the random sum $S_{\eta }=\xi _{1}+\xi _{2}+\cdots +\xi _{\eta }$ belongs to the class of consistently varying distributions. In our consideration, the random variables $\{\xi _{1},\xi _{2},\dots \}$ are not necessarily identically distributed.

Large deviation principle for one-dimensional SDEs with discontinuous coefficients

Alexei Kulik Daryna Sobolieva

https://doi.org/10.15559/16-VMSTA57

Pub. online: 1 Jul 2016 Type: Research Article

Open Access

Journal: Modern Stochastics: Theory and Applications Volume 3, Issue 2 (2016), pp. 145–164

Abstract

We establish the large deviation principle for solutions of one-dimensional SDEs with discontinuous coefficients. The main statement is formulated in a form similar to the classical Wentzel–Freidlin theorem, but under the considerably weaker assumption that the coefficients have no discontinuities of the second kind.

22 23 24 25 26

Items per page

Export citation

Copy and paste formatted citation

Formatted citation

Placeholder

Citation style

Download citation in file

Export format

Authors

Placeholder

RSS

MSTA

Online ISSN: 2351-6054
Print ISSN: 2351-6046

About

About journal
Indexed in
Editors-in-Chief

For contributors

Submit
OA Policy
Become a Peer-reviewer

Contact us

ejournals-vmsta@vtex.lt
Mokslininkų 2A
LT-08412 Vilnius
Lithuania