Stochastic differential equations of the form
are considered where is an -dimensional Brownian motion, is a right-continuous Markov chain with values in , and , satisfy suitable Itô-type conditions for the existence and uniqueness of the solution. Note that this equation can be regarded as a result of equations
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.