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In this paper, we study the global well-posedness of the stochastic S-KdVsystem in $H^1(\mathbb{R})\times H^1(\mathbb{R})$, which are driven by additivenoises. It is difficult to show the global well-posedness of a relatedperturbation system even for smooth datum and stochastic forces. To overcomeit, we introduce a new sequence of approximation equations, which is the key ofthis paper. We establish priori estimates, global well-posedness andconvergences of these approximation equations, which help us to get a pathwisepriori estimate of the initial system.

On the global well-posedness of stochastic Schr\"{o}dinger-Korteweg-de Vries system