On almost sure sample stability of nonlinear stochastic dynamic systems

In this note, an extension of Khasminskii´s theorem on almost sure stability of linear stochastic differential equations to a class of nonlinear stochastic differential equations is presented. The necessary and sufficient conditions for almost sure stability are proved. It is shown that in the second-order case, the stable region can be exactly determined by studying the singular boundaries of one-dimensional diffusion processes. The authors present a modified form of Feller´s criteria for classification of singular boundaries. The new criteria are equivalent to and much simpler for applications than Feller´s criteria. Two examples of nonlinear stochastic dynamic systems with stable regions illustrate the application procedures