New developments in almost sure sample stability of nonlinear stochastic dynamic systems

A previously published theorem (see R.Z. Khasminskii, Th. Prob. Appl.1, p.144-7, 1967) is extended to a class of nonlinear stochastic differential equations with homogeneous terms. The necessary and sufficient conditions for almost sure stability are proved for the nonlinear case. It is shown that, in the second-order case, the stable region can be exactly determined by studying the singular boundaries of a one-dimensional diffusion process. The concepts of shunt index, trap index and character value for classification of the singular boundaries are presented. As a result, a whole set of classification criteria has been developed. It is equivalent to, but much simpler than, W. Feller´s criteria (1952, 1954). Two examples of nonlinear stochastic dynamic systems with stable regions are presented