Stability of solutions of stochastic differential equations with random delays
type: Item
instance of: scholarly article
title: Stability of solutions of stochastic differential equations with random delays
zbMATH Open document ID: 1073.60068
published in: Differential Equations
publication date: 2005-06-17T00:00:00Z
review text: Stochastic functional differential equations of the form \( dx(t) = (\bar{V}x)(t)dZ(t),\) where \(\bar{V}\) is given by \((\bar{V}x)(t)=(\sum_{j=0}^{m_j}A_{1j}(t)x(h_{1j}(t)),\ldots, \sum_{j=0}^{m_m}A_{mj}(t)x(h_{mj}(t)))\), are studied. Here, \(A_{ij}\) are matrices, \(Z\) is an \(m\)-dimensional semimartingale and the memory-type arguments of \(x\) are predictable random processes. Stability (with respect to the zero solution) of solutions of these equations is defined in terms of these solutions belonging to the linear space \[ M_p^{\gamma} = \left\{ x : x \in D^n, \| x\| _{M_p^{\gamma}} := \left( \sup_{t \geq 0} E| \gamma(t) x(t)| ^p\right)^{1/p} < \infty\right\}, \] where \(D^n\) is a linear space of \(n\)-dimensional predictable stochastic processes. This notion of stability includes stability in the \(p\)th mean with decay rate \(\gamma(t)\), e.g. \(\liminf_{t \rightarrow \infty} \gamma(t) = \infty\) corresponds to asymptotic stability in the \(p\)th mean, and \(\gamma(t)= \exp(\beta t)\) to exponential stability in the \(p\)th mean. Two theorems are presented concerning \(M_{2p}^1\)- and \(M_{2p}^{1/\gamma}\)-stability of the zero solution of the above equation.
reviewed by: Evelyn Buckwar
Mathematics Subject Classification ID: 60H20
Mathematics Subject Classification ID: 34K50
Mathematics Subject Classification ID: 34F05
Mathematics Subject Classification ID: 34K20
zbMATH DE Number: 2176644
zbMATH Keywords: stochastic delay differential equations
zbMATH Keywords: stochastic stability
author: Ramazan Kadiev
MaRDI profile type: Publication
full work available at URL: b:dieq.0000033717.85317.b3
OpenAlex ID: W2083272557
DOI: 10.1023/B:DIEQ.0000033717.85317.B3
description: scientific article; zbMATH DE number 2176644