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Approximation of solutions of the stochastic wave equation by using the Fourier series

Vadym Radchenko

Nelia Stefans’ka

https://doi.org/10.15559/18-MSTA115

Pub. online: 19 Sep 2018 Type: Research Article

Open Access

Journal: Modern Stochastics: Theory and Applications* Volume 5, Issue 4 (2018), pp. 429–444

Abstract

A one-dimensional stochastic wave equation driven by a general stochastic measure is studied in this paper. The Fourier series expansion of stochastic measures is considered. It is proved that changing the integrator by the corresponding partial sums or by Fejèr sums we obtain the approximations of mild solution of the equation.

Detecting independence of random vectors: generalized distance covariance and Gaussian covariance

Björn Böttcher Martin Keller-Ressel René L. Schilling

https://doi.org/10.15559/18-VMSTA116

Pub. online: 19 Sep 2018 Type: Research Article

Open Access

Journal: Modern Stochastics: Theory and Applications Volume 5, Issue 3 (2018), pp. 353–383

Abstract

Distance covariance is a quantity to measure the dependence of two random vectors. We show that the original concept introduced and developed by Székely, Rizzo and Bakirov can be embedded into a more general framework based on symmetric Lévy measures and the corresponding real-valued continuous negative definite functions. The Lévy measures replace the weight functions used in the original definition of distance covariance. All essential properties of distance covariance are preserved in this new framework.

From a practical point of view this allows less restrictive moment conditions on the underlying random variables and one can use other distance functions than Euclidean distance, e.g. Minkowski distance. Most importantly, it serves as the basic building block for distance multivariance, a quantity to measure and estimate dependence of multiple random vectors, which is introduced in a follow-up paper [Distance Multivariance: New dependence measures for random vectors (submitted). Revised version of arXiv: 1711.07775v1] to the present article.

Detecting independence of random vectors: generalized distance covariance and Gaussian covariance

Björn Böttcher Martin Keller-Ressel René L. Schilling

https://doi.org/10.15559/18-MSTA116

Pub. online: 19 Sep 2018 Type: Research Article

Open Access

Journal: Modern Stochastics: Theory and Applications* Volume 5, Issue 3 (2018), pp. 353–383

Abstract

On a bound of the absolute constant in the Berry–Esseen inequality for i.i.d. Bernoulli random variables

Anatolii Zolotukhin Sergei Nagaev Vladimir Chebotarev

https://doi.org/10.15559/18-VMSTA113

Pub. online: 14 Sep 2018 Type: Research Article

Open Access

Journal: Modern Stochastics: Theory and Applications Volume 5, Issue 3 (2018), pp. 385–410

Abstract

It is shown that the absolute constant in the Berry–Esseen inequality for i.i.d. Bernoulli random variables is strictly less than the Esseen constant, if $1\le n\le 500000$, where n is a number of summands. This result is got both with the help of a supercomputer and an interpolation theorem, which is proved in the paper as well. In addition, applying the method developed by S. Nagaev and V. Chebotarev in 2009–2011, an upper bound is obtained for the absolute constant in the Berry–Esseen inequality in the case under consideration, which differs from the Esseen constant by no more than 0.06%. As an auxiliary result, we prove a bound in the local Moivre–Laplace theorem which has a simple and explicit form.

Despite the best possible result, obtained by J. Schulz in 2016, we propose our approach to the problem of finding the absolute constant in the Berry–Esseen inequality for two-point distributions since this approach, combining analytical methods and the use of computers, could be useful in solving other mathematical problems.

On a bound of the absolute constant in the Berry–Esseen inequality for i.i.d. Bernoulli random variables

Anatolii Zolotukhin Sergei Nagaev Vladimir Chebotarev

https://doi.org/10.15559/18-MSTA113

Pub. online: 14 Sep 2018 Type: Research Article

Open Access

Journal: Modern Stochastics: Theory and Applications* Volume 5, Issue 3 (2018), pp. 385–410

Abstract

Asymptotic arbitrage in fractional mixed markets

Fernando Cordero Irene Klein Lavinia Perez-Ostafe

https://doi.org/10.15559/18-VMSTA109

Pub. online: 20 Aug 2018 Type: Research Article

Open Access

Journal: Modern Stochastics: Theory and Applications Volume 5, Issue 4 (2018), pp. 415–428

Abstract

We consider a family of mixed processes given as the sum of a fractional Brownian motion with Hurst parameter $H\in (3/4,1)$ and a multiple of an independent standard Brownian motion, the family being indexed by the scaling factor in front of the Brownian motion. We analyze the underlying markets with methods from large financial markets. More precisely, we show the existence of a strong asymptotic arbitrage (defined as in Kabanov and Kramkov [Finance Stoch. 2(2), 143–172 (1998)]) when the scaling factor converges to zero. We apply a result of Kabanov and Kramkov [Finance Stoch. 2(2), 143–172 (1998)] that characterizes the notion of strong asymptotic arbitrage in terms of the entire asymptotic separation of two sequences of probability measures. The main part of the paper consists of proving the entire separation and is based on a dichotomy result for sequences of Gaussian measures and the concept of relative entropy.

Asymptotic arbitrage in fractional mixed markets

Fernando Cordero Irene Klein Lavinia Perez-Ostafe

https://doi.org/10.15559/18-MSTA109

Pub. online: 20 Aug 2018 Type: Research Article

Open Access

Journal: Modern Stochastics: Theory and Applications* Volume 5, Issue 4 (2018), pp. 415–428

Abstract

Multi-condition of stability for nonlinear stochastic non-autonomous delay differential equation

Leonid Shaikhet

https://doi.org/10.15559/18-VMSTA110

Pub. online: 20 Aug 2018 Type: Research Article

Open Access

Journal: Modern Stochastics: Theory and Applications Volume 5, Issue 3 (2018), pp. 337–351

Abstract

A nonlinear stochastic differential equation with the order of nonlinearity higher than one, with several discrete and distributed delays and time varying coefficients is considered. It is shown that the sufficient conditions for exponential mean square stability of the linear part of the considered nonlinear equation also are sufficient conditions for stability in probability of the initial nonlinear equation. Some new sufficient condition of stability in probability for the zero solution of the considered nonlinear non-autonomous stochastic differential equation is obtained which can be considered as a multi-condition of stability because it allows to get for one considered equation at once several different complementary of each other sufficient stability conditions. The obtained results are illustrated with numerical simulations and figures.

Multi-condition of stability for nonlinear stochastic non-autonomous delay differential equation

Leonid Shaikhet

https://doi.org/10.15559/18-MSTA110

Pub. online: 20 Aug 2018 Type: Research Article

Open Access

Journal: Modern Stochastics: Theory and Applications* Volume 5, Issue 3 (2018), pp. 337–351

Abstract

A limit theorem for a class of stationary increments Lévy moving average process with multiple singularities

Mathias Mørck Ljungdahl Mark Podolskij

https://doi.org/10.15559/18-VMSTA111

Pub. online: 20 Aug 2018 Type: Research Article

Open Access

Journal: Modern Stochastics: Theory and Applications Volume 5, Issue 3 (2018), pp. 297–316

Abstract

In this paper we present some new limit theorems for power variations of stationary increment Lévy driven moving average processes. Recently, such asymptotic results have been investigated in [Ann. Probab. 45(6B) (2017), 4477–4528, Festschrift for Bernt Øksendal, Stochastics 81(1) (2017), 360–383] under the assumption that the kernel function potentially exhibits a singular behaviour at 0. The aim of this work is to demonstrate how some of the results change when the kernel function has multiple singularity points. Our paper is also related to the article [Stoch. Process. Appl. 125(2) (2014), 653–677] that studied the same mathematical question for the class of Brownian semi-stationary models.

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