In this talk, we consider the Cauchy problem for the defocusing cubic stochastic nonlinear wave equation (SNLW) with additive space-time white noise forcing, posed in two spatial dimensions, both on the torus and on the Euclidean space. Because of the roughness of the forcing, we will use a modified version of the I-method to get global well posedness for this equation on the torus. Furthermore, we will discuss how to combine this argument with finite speed of propagation to get the same result on $\mathbb{R}^2$.