Stability preserving mappings for stochastic dynamical systems

We first formulate a general model for stochastic dynamical systems that is suitable in the stability analysis of invariant sets. This model is sufficiently general to include as special cases most of the stochastic systems considered in the literature. We then adapt several existing stability concepts to this model and we introduce the notion of stability preserving mapping of stochastic dynamical systems. Next, we establish a result which ensures that a function is a stability preserving mapping, and we use this result in proving a comparison stability theorem for general stochastic dynamical systems. We apply the comparison stability theorem in the stability analysis of dynamical systems determined by Ito differential equations.