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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

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.

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.

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-MSTA111

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

On the infinite divisibility of distributions of some inverse subordinators

Arun Kumar Erkan Nane

https://doi.org/10.15559/18-VMSTA108

Pub. online: 20 Jul 2018 Type: Research Article

Open Access

Journal: Modern Stochastics: Theory and Applications Volume 5, Issue 4 (2018), pp. 509–519

Abstract

We consider the infinite divisibility of distributions of some well-known inverse subordinators. Using a tail probability bound, we establish that distributions of many of the inverse subordinators used in the literature are not infinitely divisible. We further show that the distribution of a renewal process time-changed by an inverse stable subordinator is not infinitely divisible, which in particular implies that the distribution of the fractional Poisson process is not infinitely divisible.

On the infinite divisibility of distributions of some inverse subordinators

Arun Kumar Erkan Nane

https://doi.org/10.15559/18-MSTA108

Pub. online: 20 Jul 2018 Type: Research Article

Open Access

Journal: Modern Stochastics: Theory and Applications* Volume 5, Issue 4 (2018), pp. 509–519

Abstract

Cliquet option pricing in a jump-diffusion Lévy model

Markus Hess

https://doi.org/10.15559/18-VMSTA107

Pub. online: 20 Jul 2018 Type: Research Article

Open Access

Journal: Modern Stochastics: Theory and Applications Volume 5, Issue 3 (2018), pp. 317–336

Abstract

We investigate the pricing of cliquet options in a jump-diffusion model. The considered option is of monthly sum cap style while the underlying stock price model is driven by a drifted Lévy process entailing a Brownian diffusion component as well as compound Poisson jumps. We also derive representations for the density and distribution function of the emerging Lévy process. In this setting, we infer semi-analytic expressions for the cliquet option price by two different approaches. The first one involves the probability distribution function of the driving Lévy process whereas the second draws upon Fourier transform techniques. With view on sensitivity analysis and hedging purposes, we eventually deduce representations for several Greeks while putting emphasis on the Vega.

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