In　this　paper,　we　shall　prove　some　properties　of　set-valued　asymptotic　martingale　(amart　for　short)　and　provide　an　optional　sampling　theorem.　We　shall　also　prove　a　quasi-Riesz　decomposition　theorem　for　set-valued　amarts.　Finally　we　shall　discuss　the　existence　of　selections　of　set-valued　amarts　and　give　a　representation　theorem　of　compact　convex　set-valued　amarts.　(C)　2007　Published　by　Elsevier　Inc.