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Martingale-like sequences in Banach lattices

Haile Gessesse Alexander Melnikov

https://doi.org/10.15559/18-VMSTA120

Pub. online: 7 Nov 2018 Type: Research Article

Open Access

Journal: Modern Stochastics: Theory and Applications Volume 5, Issue 4 (2018), pp. 501–508

Abstract

Martingale-like sequences in vector lattice and Banach lattice frameworks are defined in the same way as martingales are defined in [Positivity 9 (2005), 437–456]. In these frameworks, a collection of bounded X-martingales is shown to be a Banach space under the supremum norm, and under some conditions it is also a Banach lattice with coordinate-wise order. Moreover, a necessary and sufficient condition is presented for the collection of $\mathcal{E}$-martingales to be a vector lattice with coordinate-wise order. It is also shown that the collection of bounded $\mathcal{E}$-martingales is a normed lattice but not necessarily a Banach space under the supremum norm.

Large deviations for conditionally Gaussian processes: estimates of level crossing probability

Barbara Pacchiarotti Alessandro Pigliacelli

https://doi.org/10.15559/18-VMSTA119

Pub. online: 12 Oct 2018 Type: Research Article

Open Access

Journal: Modern Stochastics: Theory and Applications Volume 5, Issue 4 (2018), pp. 483–499

Abstract

The problem of (pathwise) large deviations for conditionally continuous Gaussian processes is investigated. The theory of large deviations for Gaussian processes is extended to the wider class of random processes – the conditionally Gaussian processes. The estimates of level crossing probability for such processes are given as an application.

Ruin probability for the bi-seasonal discrete time risk model with dependent claims

Olga Navickienė Jonas Sprindys Jonas Šiaulys

https://doi.org/10.15559/18-VMSTA118

Pub. online: 1 Oct 2018 Type: Research Article

Open Access

Journal: Modern Stochastics: Theory and Applications Volume 6, Issue 1 (2019), pp. 133–144

Abstract

The discrete time risk model with two seasons and dependent claims is considered. An algorithm is created for computing the values of the ultimate ruin probability. Theoretical results are illustrated with numerical examples.

On generalized stochastic fractional integrals and related inequalities

Hüseyin Budak Mehmet Zeki Sarikaya

https://doi.org/10.15559/18-VMSTA117

Pub. online: 24 Sep 2018 Type: Research Article

Open Access

Journal: Modern Stochastics: Theory and Applications Volume 5, Issue 4 (2018), pp. 471–481

Abstract

The generalized mean-square fractional integrals ${\mathcal{J}_{\rho ,\lambda ,u+;\omega }^{\sigma }}$ and ${\mathcal{J}_{\rho ,\lambda ,v-;\omega }^{\sigma }}$ of the stochastic process X are introduced. Then, for Jensen-convex and strongly convex stochastic proceses, the generalized fractional Hermite–Hadamard inequality is establish via generalized stochastic fractional integrals.

Stochastic models associated to a Nonlocal Porous Medium Equation

Alessandro De Gregorio

https://doi.org/10.15559/18-VMSTA112

Pub. online: 19 Sep 2018 Type: Research Article

Open Access

Journal: Modern Stochastics: Theory and Applications Volume 5, Issue 4 (2018), pp. 457–470

Abstract

The nonlocal porous medium equation considered in this paper is a degenerate nonlinear evolution equation involving a space pseudo-differential operator of fractional order. This space-fractional equation admits an explicit, nonnegative, compactly supported weak solution representing a probability density function. In this paper we analyze the link between isotropic transport processes, or random flights, and the nonlocal porous medium equation. In particular, we focus our attention on the interpretation of the weak solution of the nonlinear diffusion equation by means of random flights.

Drifted Brownian motions governed by fractional tempered derivatives

Mirko D’Ovidio Francesco Iafrate Enzo Orsingher

https://doi.org/10.15559/18-VMSTA114

Pub. online: 19 Sep 2018 Type: Research Article

Open Access

Journal: Modern Stochastics: Theory and Applications Volume 5, Issue 4 (2018), pp. 445–456

Abstract

Fractional equations governing the distribution of reflecting drifted Brownian motions are presented. The equations are expressed in terms of tempered Riemann–Liouville type derivatives. For these operators a Marchaud-type form is obtained and a Riesz tempered fractional derivative is examined, together with its Fourier transform.

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

Vadym Radchenko

Nelia Stefans’ka

https://doi.org/10.15559/18-VMSTA115

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.

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.

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.

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