Algebraic stability of non-homogeneous diffusion processes with and without regime-switching

Some sufficient conditions on the algebraic stability of non-homogeneous diffusion processes and regime-switching diffusion processes are established. In this work we focus on determining the decay rate of a stochastic system which switches randomly between different states, and owns different decay rates at various states. In particular, we show that if a two-state regime-switching diffusion process is p-th moment exponentially stable in one state and is p-th moment algebraically stable in another state, which are characterized by a common Lyapunov function, then this process is ultimately exponentially stable regardless of the jumping rate of the random switching between two states.