On the partial stability in probability of nonlinear stochastic systems

A general class of the nonlinear time-varying systems of Itô stochastic differential equations is considered. Two problems on the partial stability in probability are studied as follows: 1) the stability with respect to a given part of the variables of the trivial equilibrium; 2) the stability with respect to a given part of the variables of the partial equilibrium. The stochastic Lyapunov functions-based conditions of the partial stability in probability are established. In addition to the main Lyapunov function, an auxiliary (generally speaking, vector-valued) function is introduced for correcting the domain in which the main Lyapunov function is constructed. A comparison with the well-known results on the partial stability of the systems of stochastic differential equations is given. An example that well illustrates the peculiarities of the suggested approach is described. Also a possible unified approach to analyze the partial stability of the time-invariant and time-varying systems of stochastic differential equations is discussed.