Stochastic Functional Differential Equation under Regime Switching

We discuss stochastic functional differential equation under regime switching dx(t)=f(xt,r(t),t)dt+q(r(t))x(t)dW1(t)+σ(r(t))|x(t)|βx(t)dW2(t). We obtain unique global solution of this system without the linear growth condition; furthermore, we prove its asymptotic ultimate boundedness. Using the ergodic property of the Markov chain, we give the sufficient condition of almost surely exponentially stable of this system