  We study the stochastic Nonlinear Schr\"{o}dinger system with multiplicative
white noise in energy space $H^1$. Based on deterministic and stochastic
Strichartz estimates, we prove the local well-posedness and uniqueness of mild
solution. Then we prove the global well-posedness in the mass subcritical case
and the defocusing case. For the mass subcritical case, we also investigate the
global existence when the $L^2$ norm of initial value is small enough. In
addition, we study the blow-up phenomenon and give a sharp criteria.
