Practical Stability in thepth Mean for Itô Stochastic Differential Equations

Abstract

Thepth mean practical stability problem is studied for a general class of Itô-type stochastic differential equations over both finite and infinite time horizons. Instead of the comparison principle, a functionη(t)which is nonnegative, nondecreasing, and differentiable is cooperated with the Lyapunov-like functions to analyze the practical stability. By using this technique, the difficulty in finding an auxiliary deterministic stable system is avoided. Then, some sufficient conditions are established that guarantee thepth moment practical stability of the considered equations. Moreover, the practical stability is compared with traditional Lyapunov stability; some differences between them are given. Finally, the results derived in this paper are demonstrated by an illustrative example.