  In this paper, we present a globalization argument for stochastic nonlinear
dispersive PDEs with additive noises by adapting the $I$-method (= the method
of almost conservation laws) to the stochastic setting. As a model example, we
consider the defocusing stochastic cubic nonlinear Schr\"odinger equation
(SNLS) on $\mathbb R^3$ with additive stochastic forcing, white in time and
correlated in space, such that the noise lies below the energy space. By
combining the $I$-method with Ito's lemma and a stopping time argument, we
construct global-in-time dynamics for SNLS below the energy space.
