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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.

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-MSTA69

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

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.

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

A X = B

Alexander Kukush Yaroslav Tsaregorodtsev

https://doi.org/10.15559/16-MSTA67

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

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.

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-MSTA66

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

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

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).

Workshop “Fractality and Fractionality”

Yuliya Mishura Georgiy Shevchenko

https://doi.org/10.15559/16-MSTA65

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.

Averaged deviations of Orlicz processes and majorizing measures

Rostyslav Yamnenko

https://doi.org/10.15559/16-MSTA64

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

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