A note on the LaSalle-type theorems for stochastic differential delay equations
type: Item
instance of: scholarly article
title: A note on the LaSalle-type theorems for stochastic differential delay equations
zbMATH Open document ID: 0996.60064
author: Xuerong Mao
published in: Journal of Mathematical Analysis and Applications
publication date: 2002-07-04T00:00:00Z
review text: The author presents an improvement of results obtained in an earlier paper [J. Math. Anal. Appl. 236, No. 2, 350-369 (1999; Zbl 0958.60057)]. In the article \(n\)-dimensional stochastic differential delay equations are considered, which are of the form \[ dx(t)=f(x(t),x(t-\tau),t) dt + g(x(t),x(t-\tau),t) dB(t),\tag{1} \] where \(B(t)\) denotes \(m\)-dimensional Brownian motion. The main theorem is a stochastic version of the LaSalle theorem, providing criteria for the determination of the almost sure asymptotic behaviour of the solution of (1). The improvement concerns the assumptions on the coefficient functions. The local Lipschitz and local linear growth conditions on \(f\) and \(g\) are relaxed to local boundedness in the first two arguments and uniform boundedness in the last argument of \(f\) and \(g\), in addition the existence and uniqueness of a solution of (1) is required. The results can thus be applied to a larger class of equations. The proof of the theorem, some corollaries and an extension to the multiple delay case are given. Several examples are presented, demonstrating the usefulness of the results.
reviewed by: Evelyn Buckwar
Mathematics Subject Classification ID: 60H10
Mathematics Subject Classification ID: 34K50
Mathematics Subject Classification ID: 93D05
zbMATH DE Number: 1763448
zbMATH Keywords: stochastic delay differential equations
zbMATH Keywords: stochastic stability
zbMATH Keywords: Lyapunov stability
zbMATH Keywords: LaSalle theorem
MaRDI profile type: Publication
OpenAlex ID: W2125968149
cites work: Introduction to functional differential equations
cites work: Stability theory for ordinary differential equations
cites work: Q3995465
cites work: Q4338975
cites work: LaSalle-type theorems for stochastic differential delay equations
cites work: Robustness of exponential stability of stochastic differential delay equations
DOI: 10.1006/JMAA.2001.7803
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description: scientific article; zbMATH DE number 1763448