Stability of hybrid stochastic delay systems whose discrete components have a large state space: a two-time-scale approach
The Itô delay stochastic differential equation \(dx(t)=f(x(t),x(t-\tau),r(t))\,dt+g(x(t),x(t-\tau),r(t))\,dW(t)\) on \(\mathbb R^n\) with \(t\mapsto r(t)\) a Markov chain on a large finite state space \(S\), independent of the Wiener process \(W\), and \(f(0,0,r)=g(0,0,r)=0\) for all \(r\in S\) is considered. Assuming the state space \(S\) to be decomposable into subsets such that changes of \(r\) between these subsets occur only rarely one may consider a Markov chain on a smaller state space as a limiting case. Conditions are given for \(p\)th moment exponential stability of the limiting system to imply the same sort of stability for the original system. Furthermore, almost sure stability and stability in distribution are discussed.
scientific article; zbMATH DE number 5710072
scientific article