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

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

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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.

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.

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

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

BAŠTINEC, J.; KLIMEŠOVÁ, M.

2017

21.12.2016

University of Defence

Brno

978-80-7231-400-3

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

7

30

111

http://mitav.unob.cz/data/MITAV%202016%20Proceedings.pdf