右连续上鞅的提升及其应用

THE LIFTING OF RIGHT CONTINUOUS SUPERMARTINGALE WITH ITS APPLICATION

本文首先给出了一个右连续上鞅的ＳＤ提升，在引进Ｓ－上鞅和强Ｓ－上鞅概念之后，研究了一致可积上鞅与Ｓ－上鞅，类（Ｄ）上鞅与强Ｓ－上鞅之间的关系，并得到了Ｓ－上鞅与强Ｓ－上鞅的许多性质，作为其直接结果，给出了类（Ｄ）上鞅的Ｄｏｏｂ－Ｍｅｙｅｒ分解．最后讨论了一个一致可积上鞅及其ＳＤ提升分别决定的Ｄｏｌｅａｎｓ测度之间的关系，给出了一致可积上鞅所决定的Ｄｏｌｅａｎｓ测度的Ｌｏｅｂ表示，并由此证明，在一个Ｌｏｅｂ概率空间中，右连续上鞅一致可积当且仅当它是类（Ｄ）

In the paper, the SD lifting of right continuous supermartingale is obtained. By introducing S supermartingale and strong S supermartingale, the relations between u.i.supermartingale and S supermartinale, class (D) supermartingale and strong S supermartingale are discussed respectively. And many properties about S supermartingale and strong S supermartingale are obtained, which lead to the Doob Meyer decomposition of class (D) supermartingale. Finally, the relations between the Doleans measures, which are defined by u.i. supermartingale and its SD lifting respectively, are discussed. The Loeb representation of the Doleans measure generated by u.i. supermartingale is obtained. The equivalence between u.i.supermartingale and class (D) supermartingale in the Loeb probability space is proved.