In　a　separable　Banach　space,　for　set-valued　martingale.　several　equivalent　conditions　based　on　the　measurable　selections　are　discussed,　and　then,　in　an　M-type　2　Banach　space,　at　first　we　define　single　valued　stochastic　integral　by　the　differential　of　a　real　valued　Brownian　motion,　after　that　extend　it　to　set-valued　case.　We　prove　that　the　set-valued　stochastic　integral　becomes　a　set-valued　submartingale,　which　is　different　from　single　valued　case,　and　obtain　the　Castaing　representation　theorem　for　the　set-valued　stochastic　integral,　which　is　applicable　for　set-valued　stochastic　differential　equations.　(c)　2008　Elsevier　Inc.　All　rights　reserved.