Stability of stochastic differential equations with Markovian switching
type: Item
instance of: scholarly article
title: Stability of stochastic differential equations with Markovian switching
zbMATH Open document ID: 0962.60043
DOI: 10.1016/S0304-4149(98)00070-2
author: Xuerong Mao
published in: Stochastic Processes and their Applications
publication date: 2001-01-17T00:00:00Z
review text: Stochastic differential equations of the form \[ dx(t)=f(x(t),t,r(t)) dt + g(x(t),t,r(t)) dw(t) \] are considered where \(w(t)\) is an \(m\)-dimensional Brownian motion, \(r(t)\) is a right-continuous Markov chain with values in \(S:=\{1,2,\dots{},N\}\), and \(f:\mathbb R^n\times \mathbb R_+\times S\to \mathbb R^n\), \(g:\mathbb R^n\times \mathbb R_+\times S\to \mathbb R^{n\times m}\) satisfy suitable Itô-type conditions for the existence and uniqueness of the solution. Note that this equation can be regarded as a result of \(N\) equations \[ dx(t)=f(x(t),t,i) dt + g(x(t),t,i) dw(t),\quad 1\leq i\leq N, \] switching from one to other according to the movement of the Markov chain. Criteria for exponential stability of the moments and for a.s. exponential stability are given, special attention being devoted to the linear equations and to nonlinear deterministic jump equations.
Mathematics Subject Classification ID: 60H10
zbMATH DE Number: 1554273
zbMATH Keywords: Lyapunov exponent
zbMATH Keywords: Markovian switching
zbMATH Keywords: Brownian motion
MaRDI profile type: Publication
cites work: Stability of a random diffusion with linear drift
cites work: Q4326384
cites work: Q4086525
cites work: Controllability, stabilizability, and continuous-time Markovian jump linear quadratic control
cites work: Q4229393
cites work: Random differential inequalities
cites work: Q3999821
cites work: Q4303533
cites work: Q4369402
cites work: Q4693053
description: scientific article; zbMATH DE number 1554273