Stochastic functional differential equation under regime switching (Q1925474)

Summary: We discuss the stochastic functional differential equation under regime switching \[ dx(t) = f(x_t, r(t), t)dt + q(r(t))x(t)dW_1(t) + \sigma(r(t))|x(t)|^\beta x(t)dW_2(t). \] We obtain a 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.