Asymptotic uniform stability in probability of nontrivial solution of nonlinear stochastic systems

The aim of this paper is to study the asymptotic uniform stability in probability when a nonlinear stochastic differential equation does not have a trivial solution. For nontrivial solutions of a nonlinear stochastic differential equation, the problem of asymptotic uniform stability in probability is reformulated for a ball of radius R>0. Based on this new formulation, a theorem for the asymptotic uniform stability in probability for this ball is proposed by using a Lyapunov approach.