Publication detail

Stability of the Zero Solution of Stochastic Differential Systems with Four-Dimensional Brownian Motion

BAŠTINEC, J. KLIMEŠOVÁ, M.

Original Title

Stability of the Zero Solution of Stochastic Differential Systems with Four-Dimensional Brownian Motion

English Title

Stability of the Zero Solution of Stochastic Differential Systems with Four-Dimensional Brownian Motion

Type

conference paper

Language

en

Original Abstract

The natural world is influenced by stochasticity therefore stochastic models are used to test various situations because only the stochastic model can approximate the real model. For example, the stochastic model is used in population, epidemic and genetic simulations in medicine and biology, for simulations in physical and technical sciences, for analysis in economy, financial mathematics, etc. The crucial characteristic of the stochastic model is its stability. Stability of stochastic differential equations (SDEs) has become a very popular theme of recent research in mathematics and its applications. This article studies the fundamental theory of the stochastic stability. There is investigated the stability of the solution of stochastic differential equations and systems of SDEs. The article begins with a summary of the stochastic theory. Then, there are inferred conditions for the asymptotic mean square stability of the zero solution of stochastic system with Brownian motion. There is used a Lyapunov function for proofs of main results. The method of Lyapunov functions for the analysis of qualitative behavior of SDEs provides some very useful information in the study of stability properties for concrete stochastic dynamical systems, conditions of existence the stationary solutions of SDEs and related problems. There are proved conditions for the stability (asymptotic, stochastic asymptotic). The results are illustrated by trivial examples for special types of matrices.

English abstract

The natural world is influenced by stochasticity therefore stochastic models are used to test various situations because only the stochastic model can approximate the real model. For example, the stochastic model is used in population, epidemic and genetic simulations in medicine and biology, for simulations in physical and technical sciences, for analysis in economy, financial mathematics, etc. The crucial characteristic of the stochastic model is its stability. Stability of stochastic differential equations (SDEs) has become a very popular theme of recent research in mathematics and its applications. This article studies the fundamental theory of the stochastic stability. There is investigated the stability of the solution of stochastic differential equations and systems of SDEs. The article begins with a summary of the stochastic theory. Then, there are inferred conditions for the asymptotic mean square stability of the zero solution of stochastic system with Brownian motion. There is used a Lyapunov function for proofs of main results. The method of Lyapunov functions for the analysis of qualitative behavior of SDEs provides some very useful information in the study of stability properties for concrete stochastic dynamical systems, conditions of existence the stationary solutions of SDEs and related problems. There are proved conditions for the stability (asymptotic, stochastic asymptotic). The results are illustrated by trivial examples for special types of matrices.

Keywords

Brownian motion; stochastic differential equation; Lyapunov function; stochastic Lyapunov function; stability; stochastic stability.

Released

21.12.2016

Publisher

University of Defence

Location

Brno

ISBN

978-80-7231-400-3

Book

Mathematics, Information Technologies and Applied Sciences 2016 (post-conference proceedings of extended versions of selected papers )

Pages from

7

Pages to

30

Pages count

111

URL

Documents

BibTex


@inproceedings{BUT131309,
  author="Marie {Klimešová} and Jaromír {Baštinec}",
  title="Stability of the Zero Solution of Stochastic Differential Systems with Four-Dimensional
Brownian Motion",
  annote="The natural world is influenced by stochasticity therefore stochastic
models are used to test various situations because only the stochastic model can
approximate the real model. For example, the stochastic model is used in population,
epidemic and genetic simulations in medicine and biology, for simulations
in physical and technical sciences, for analysis in economy, financial mathematics,
etc. The crucial characteristic of the stochastic model is its stability. Stability
of stochastic differential equations (SDEs) has become a very popular theme of
recent research in mathematics and its applications. This article studies the fundamental
theory of the stochastic stability. There is investigated the stability of
the solution of stochastic differential equations and systems of SDEs. The article
begins with a summary of the stochastic theory. Then, there are inferred conditions
for the asymptotic mean square stability of the zero solution of stochastic
system with Brownian motion. There is used a Lyapunov function for proofs of
main results. The method of Lyapunov functions for the analysis of qualitative
behavior of SDEs provides some very useful information in the study of stability
properties for concrete stochastic dynamical systems, conditions of existence the
stationary solutions of SDEs and related problems. There are proved conditions
for the stability (asymptotic, stochastic asymptotic). The results are illustrated by
trivial examples for special types of matrices.",
  address="University of Defence",
  booktitle="Mathematics, Information Technologies and Applied Sciences 2016 (post-conference proceedings of extended versions of selected papers )",
  chapter="131309",
  howpublished="online",
  institution="University of Defence",
  year="2016",
  month="december",
  pages="7--30",
  publisher="University of Defence",
  type="conference paper"
}