无穷可数个Brown运动驱动的随机微分方程解的分布唯一性及路径唯一性(英文)

On the Uniqueness in Law and the Pathwise Uniqueness for a SDE Driven by Countably Many Brownian Motions

研究如下形式的随机微分方程Xti=xi+∑∞j=∫10tσsij(Xs)dBjs+∫0tbis(Xs)ds,i=1,2,…,n,(*)其中{Btj}j∞=1是相互独立的标准Brown运动的无穷可数序列.主要证明如下结论:1)解的分布唯一性蕴含了解的联合分布唯一性;2)解的分布唯一性与强解的存在性可以保证解的轨道唯一性.结论2)是Yamada定理的对偶命题.

Consider the n-dimensional SDE Xii=xi＋∞∑j=1∫l0σs^ij（X6）dBs^j＋∫10bs^1（X5）ds,i=1,2…,n,（＊）where{ Btj}j∞=1is an infinite sequence of independent standard Brownian motions. In this paper, we prove that the uniqueness in law for （ ＊ ） implies the uniqueness of the joint distribution of a pair （X,B）, and moreover we prove that the uniqueness in law for （ ＊ ） together with the strong existence,guarantees the pathwise uniqueness.