It is worth pointing out that some stochastic systems are indeed stable but subject to a certain lower decay rate which is different from exponential decay,such as polynomial or logarithmic. For more accurate quantitative analyses of stability properties,this paper extends the usual exponential stability concepts to a more general stable decay function and investigates the general decay stability of stochastic differential equations with Markovian switching. Firstly,some φ( t) -stability criteria in p-th moment and almost surely sense for the analytical solutions are established,by utilizing ItÔ formula,Borel-Cantelli and martingale exponential inequalities. Then the Euler Maruyama method is shown to be effective in capturing φ( t) -stability behavior for all sufficiently small timesteps under appropriate conditions.