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Modern Stochastics: Theory and Applications

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Journals: Modern Stochastics: Theory and Applications

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rok rok

Pub. online: 5 Nov 2024 Type: Article

Open Access

Journal: Modern Stochastics: Theory and Applications Volume 99 (2024): Test numeris, pp. 1–2

Abstract

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Pub. online: 5 Nov 2024 Type: Article

Open Access

Journal: Modern Stochastics: Theory and Applications Volume 999 (2024), pp. 1–2

Testas

Pub. online: 18 Aug 2023 Type: Article

Open Access

Journal: Modern Stochastics: Theory and Applications Volume 1, Issue 2 (2014), pp. 1–100

The Burgers equation driven by a stochastic measure

Vadym Radchenko

https://doi.org/10.15559/23-VMSTA224

Pub. online: 21 Feb 2023 Type: Research Article

Open Access

Journal: Modern Stochastics: Theory and Applications

Abstract

The class of one-dimensional equations driven by a stochastic measure μ is studied. For μ only σ-additivity in probability is assumed. This class of equations includes the Burgers equation and the heat equation. The existence and uniqueness of the solution are proved, and the averaging principle for the equation is studied.

Existence and uniqueness of weak solution to a three-dimensional stochastic modified-Leray-alpha model of fluid turbulence

Ridha Selmi Rim Nasfi

https://doi.org/10.15559/21-VMSTA175

Pub. online: 16 Mar 2021 Type: Research Article

Open Access

Journal: Modern Stochastics: Theory and Applications Volume 8, Issue 1 (2023), pp. 115–137

Abstract

In this paper, we study the stochastic three-dimensional modified Leray-alpha model arising from the turbulent flows of fluids. We prove the existence of the probabilistic weak solution under the non-Lipschitz condition for the nonlinear forcing terms. We also discuss its uniqueness.

Note on local mixing techniques for stochastic differential equations

Alexander Veretennikov

https://doi.org/10.15559/21-VMSTA174

Pub. online: 16 Mar 2021 Type: Research Article

Open Access

Journal: Modern Stochastics: Theory and Applications Volume 8, Issue 1 (2023), pp. 1–15

Abstract

The paper discusses several techniques which may be used for applying the coupling method to solutions of stochastic differential equations (SDEs). The coupling techniques traditionally consist of two components: one is local mixing, the other is recurrence. Often in the articles they do not split. Yet, they are quite different in their nature, and this paper separates them, concentrating only on the former.

Most of the techniques discussed here work in dimension $d\ge 1$, although, in $d=1$ there is one additional option to use intersections of trajectories, which requires nothing but the strong Markov property and nondegeneracy of the diffusion coefficient. In dimensions $d>1$ it is possible to use embedded Markov chains either by considering discrete times $n=0,1,\dots $, or by arranging special stopping time sequences and to use the local Markov–Dobrushin (MD) condition, which is one of the most efficient versions of local mixing. Further applications may be based on one or another version of the MD condition; respectively, this paper is devoted to various methods of verifying one or another form of it.

Long-time behavior of a nonautonomous stochastic predator–prey model with jumps

Olga Borysenko Oleksandr Borysenko

https://doi.org/10.15559/21-VMSTA173

Pub. online: 8 Mar 2021 Type: Research Article

Open Access

Journal: Modern Stochastics: Theory and Applications Volume 8, Issue 1 (2023), pp. 17–39

Abstract

The existence and uniqueness of a global positive solution is proven for the system of stochastic differential equations describing a nonautonomous stochastic predator–prey model with a modified version of the Leslie–Gower term and Holling-type II functional response disturbed by white noise, centered and noncentered Poisson noises. Sufficient conditions are obtained for stochastic ultimate boundedness, stochastic permanence, nonpersistence in the mean, weak persistence in the mean and extinction of a solution to the considered system.

Investigation of sample paths properties for some classes of φ-sub-Gaussian stochastic processes

Olha Hopkalo Lyudmyla Sakhno

https://doi.org/10.15559/21-VMSTA171

Pub. online: 26 Jan 2021 Type: Research Article

Open Access

Journal: Modern Stochastics: Theory and Applications Volume 8, Issue 1 (2023), pp. 41–62

Abstract

This paper investigates sample paths properties of φ-sub-Gaussian processes by means of entropy methods. Basing on a particular entropy integral, we treat the questions on continuity and the rate of growth of sample paths. The obtained results are then used to investigate the sample paths properties for a particular class of φ-sub-Gaussian processes related to the random heat equation. We derive the estimates for the distribution of suprema of such processes and evaluate their rate of growth.

Subject index

https://doi.org/10.15559/19-VMSTA64SI

Pub. online: 12 Dec 2019 Type: Subject Index

Open Access

Journal: Modern Stochastics: Theory and Applications Volume 6, Issue 4 (2019), pp. 523–527

2010 Mathematics Subject Classification index

https://doi.org/10.15559/19-VMSTA64MI

Pub. online: 12 Dec 2019 Type: 2010 Mathematics Subject Classification Index

Open Access

Journal: Modern Stochastics: Theory and Applications Volume 6, Issue 4 (2019), pp. 519–522

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