Bulletin of the
Korean Mathematical Society
BKMS

ISSN(Print) 1015-8634 ISSN(Online) 2234-3016

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Bull. Korean Math. Soc. 2014; 51(1): 221-227

Printed January 1, 2014

https://doi.org/10.4134/BKMS.2014.51.1.221

Copyright © The Korean Mathematical Society.

A note on exponential almost sure stability of stochastic differential equation

Xuerong Mao, Qingshuo Song, and Dichuan Yang

University of Strathclyde, City University of Hong Kong, City University of Hong Kong

Abstract

Our goal is to relax a sufficient condition for the exponential almost sure stability of a certain class of stochastic differential equations. Compared to the existing theory, we prove the almost sure stability, replacing Lipschitz continuity and linear growth conditions by the existence of a strong solution of the underlying stochastic differential equation. This result is extendable for the regime-switching system. An explicit example is provided for the illustration purpose.

Keywords: almost sure stability, stochastic differential equation, regime-switching, Besel squared process

MSC numbers: 60G52, 60G17