For stochastic systems, this article develops a Lyapunov-type characterization of integral input-to-state stability (iISS), which is truly parallel to the deterministic one. In contrast to preceding results, this article allows stochastic noise to be active globally in the state space. Deriving pathwise bounds of stochastic processes is the original key idea and it leads to new techniques to unify Lyapunov-type characterization of iISS with that of input-to-state stability (ISS). The ISS test becomes a special case of the iISS test completely, which forms a stochastic counterpart of the popular iISS/ISS framework for deterministic nonlinear systems analysis and design. The usefulness of the developed iISS/ISS characterizations is substantiated by utilizing them to consolidate stochastic versions of small-gain criteria to establish stability of interconnected systems without restricting noise effects to bounded domains. The framework of integral noise-to-state stability is also made useful by allowing the global stochastic noise.