A Lyapunov approach to iISS and iNSS for stochastic systems in path-wise probability

For stochastic nonlinear dynamical systems, this paper addresses several stability notions in the path-wise sense. Lyapunov characterizations are developed to verify those properties. The notions investigated here are stricter than boundedness properties ensured by popular input-to-state stability and noise-to-state stability whose probability is taken at each instant. It is demonstrated that Lyapunov functions constructed frequently in the literature not only establish the instantaneous probabilistic properties, but also path-wise probabilistic ones. Differences appear when one wants to estimate the convergence of initial effect. The differences disappear if diffusion fields are remotely orthogonal to the gradient of Lyapunov functions.