关于集值上鞅Doob分解的注记

Remarks on Doob Decomposition of the Set-valued Supermartingle

假定(X,‖.‖)为可分的Banach空间,X*为其对偶空间且可分.给出了集值上鞅一种新形式的Doob分解定义,证明了一维实空间集值上鞅具有这种形式的Doob分解,举例说明在二维实空间,并非集值上鞅都具有这种形式Doob分解.最后,给出了实Banach空间集值上鞅具有这种形式的Doob分解的充分必要条件.

Let（X,‖·‖） denote a real separable Banach space,whose dual space is X＊,and X＊ is also separable.First of all,a new definition of Doob decomposition for set-valued supermartingale is presented.The definition is feasible for set-valued supermartingale in one dimension real space,but it is not always feasible in two dimensions real space.In order to prove this point,some examples are given.At last,the sufficient and necessary condition of this new Doob decomposition for set-valued supermartingale in real Banach space is discussed.