On the solution of the problem of stochastic stability of the integral manifold by the Lyapunov’s second method

Abstract

By using the method of Lyapunov functions, we establish sufficient conditions of stability and asymptotic stability in probability for the integral manifold of the Itˆo differential equations in the presence of random perturbations from the class of processes with independent increments. Theorems on the stochastic stability of the analytically given integral manifold of differential equations are proved in the first approximation and under the permanent action of small (in the mean) random perturbations.