Stability of stochastic differential equations with Markovian switching. (English) Zbl 0962.60043

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.


60H10 Stochastic ordinary differential equations (aspects of stochastic analysis)
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