Dynamics of a mean-reverting stochastic volatility equation with regime switching

•We show the existence of global positive solution to Eq. (2) for any initial value, which extends the corresponding result in Mao et al. [10].•We gives the pth moment estimation, asymptotically bounded in pth moment and the Lyapunov exponent of the solution Xt to Eq. (2).•We present some sufficient conditions for that the process (Xt, rt) determined by Eq. (2) is positive recurrent and admits a unique ergodic asymptotically invariant distribution.