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:

Transport equation driven by a stochastic measure

Vadym Radchenko

https://doi.org/10.15559/23-VMSTA222

Pub. online: 6 Feb 2023 Type: Research Article

Open Access

Journal: Modern Stochastics: Theory and Applications Volume 10, Issue 2 (2023), pp. 197–209

Abstract

The stochastic transport equation is considered where the randomness is given by a symmetric integral with respect to a stochastic measure. For a stochastic measure, only σ-additivity in probability and continuity of paths is assumed. Existence and uniqueness of a weak solution to the equation are proved.

Parameter estimation in mixed fractional stochastic heat equation

Diana Avetisian Kostiantyn Ralchenko

https://doi.org/10.15559/23-VMSTA221

Pub. online: 24 Jan 2023 Type: Research Article

Open Access

Journal: Modern Stochastics: Theory and Applications Volume 10, Issue 2 (2023), pp. 175–195

Abstract

The paper is devoted to a stochastic heat equation with a mixed fractional Brownian noise. We investigate the covariance structure, stationarity, upper bounds and asymptotic behavior of the solution. Based on its discrete-time observations, we construct a strongly consistent estimator for the Hurst index H and prove the asymptotic normality for $H < 3/4$. Then assuming the parameter H to be known, we deal with joint estimation of the coefficients at the Wiener process and at the fractional Brownian motion. The quality of estimators is illustrated by simulation experiments.

A modified Φ-Sobolev inequality for canonical Lévy processes and its applications

Noriyoshi Sakuma

Ryoichi Suzuki

https://doi.org/10.15559/23-VMSTA220

Pub. online: 23 Jan 2023 Type: Research Article

Open Access

Journal: Modern Stochastics: Theory and Applications Volume 10, Issue 2 (2023), pp. 145–173

Abstract

A new modified Φ-Sobolev inequality for canonical ${L^{2}}$-Lévy processes, which are hybrid cases of the Brownian motion and pure jump-Lévy processes, is developed. Existing results included only a part of the Brownian motion process and pure jump processes. A generalized version of the Φ-Sobolev inequality for the Poisson and Wiener spaces is derived. Furthermore, the theorem can be applied to obtain concentration inequalities for canonical Lévy processes. In contrast to the measure concentration inequalities for the Brownian motion alone or pure jump Lévy processes alone, the measure concentration inequalities for canonical Lévy processes involve Lambert’s W-function. Examples of inequalities are also presented, such as the supremum of Lévy processes in the case of mixed Brownian motion and Poisson processes.

Some examples of noncentral moderate deviations for sequences of real random variables

Rita Giuliano

Claudio Macci

https://doi.org/10.15559/23-VMSTA219

Pub. online: 19 Jan 2023 Type: Research Article

Open Access

Journal: Modern Stochastics: Theory and Applications Volume 10, Issue 2 (2023), pp. 111–144

Abstract

The term moderate deviations is often used in the literature to mean a class of large deviation principles that, in some sense, fills the gap between a convergence in probability to zero (governed by a large deviation principle) and a weak convergence to a centered normal distribution. In this paper, some examples of classes of large deviation principles of this kind are presented, but the involved random variables converge weakly to Gumbel, exponential and Laplace distributions.

Reflected generalized discontinuous BSDEs with rcll barrier and an obstacle problem of IPDE with nonlinear Neumann boundary conditions

Mohammed Elhachemy

Mohamed El Otmani

https://doi.org/10.15559/22-VMSTA218

Pub. online: 30 Dec 2022 Type: Research Article

Open Access

Journal: Modern Stochastics: Theory and Applications Volume 10, Issue 1 (2023), pp. 77–110

Abstract

Reflected generalized backward stochastic differential equations (BSDEs) with one discontinuous barrier are investigated when the noise is driven by a Brownian motion and an independent Poisson measure. The existence and uniqueness of the solution are derived when the generators are monotone and the barrier is right-continuous with left limits (rcll). The link is established between this solution and a viscosity solution for an obstacle problem of integral-partial differential equations with nonlinear Neumann boundary conditions.

Lévy processes conditioned to stay in a half-space with applications to directional extremes

Jevgenijs Ivanovs

Jakob D. Thøstesen

https://doi.org/10.15559/22-VMSTA217

Pub. online: 25 Nov 2022 Type: Research Article

Open Access

Journal: Modern Stochastics: Theory and Applications Volume 10, Issue 1 (2023), pp. 59–75

Abstract

This paper provides a multivariate extension of Bertoin’s pathwise construction of a Lévy process conditioned to stay positive or negative. Thus obtained processes conditioned to stay in half-spaces are closely related to the original process on a compact time interval seen from its directional extremal points. In the case of a correlated Brownian motion the law of the conditioned process is obtained by a linear transformation of a standard Brownian motion and an independent Bessel-3 process. Further motivation is provided by a limit theorem corresponding to zooming in on a Lévy process with a Brownian part at the point of its directional infimum. Applications to zooming in at the point furthest from the origin are envisaged.

A class of fractional Ornstein–Uhlenbeck processes mixed with a Gamma distribution

Luigi Amedeo Bianchi

Stefano Bonaccorsi

Luciano Tubaro

https://doi.org/10.15559/22-VMSTA216

Pub. online: 21 Nov 2022 Type: Research Article

Open Access

Journal: Modern Stochastics: Theory and Applications Volume 10, Issue 1 (2023), pp. 37–57

Abstract

We consider a sequence of fractional Ornstein–Uhlenbeck processes, that are defined as solutions of a family of stochastic Volterra equations with a kernel given by the Riesz derivative kernel, and leading coefficients given by a sequence of independent Gamma random variables. We construct a new process by taking the empirical mean of this sequence. In our framework, the processes involved are not Markovian, hence the analysis of their asymptotic behaviour requires some ad hoc construction. In our main result, we prove the almost sure convergence in the space of trajectories of the empirical means to a given Gaussian process, which we characterize completely.

2010 Mathematics Subject Classification index

https://doi.org/10.15559/22-VMSTA94MI

Pub. online: 9 Nov 2022 Type: 2010 Mathematics Subject Classification Index

Open Access

Journal: Modern Stochastics: Theory and Applications Volume 9, Issue 4 (2022), pp. 503–506