Brown运动关于球的末遇时首出时及停留时的矩

MOMENTS OF LAST EXIT TIME, FIRST EXIT TIME AND OCCUPATION TIME FOR A BALL BY BROWNIAN MOTION

设Ｘ＝（Ｘｔ）ｔ＞Ｏ为Ｒｄ中的标准Ｂｒｏｗｎ运动，Ｂｒ＝为以原点为中心，半径为ｒ的球．分别用Ｌｒ，ｒ及Ｔｒ表示Ｘ末遇Ｂｒ，首出Ｂｒ及停留在Ｂｒ中的时间．本文给出了求这些变量的矩的一般公式，并求出了前二阶矩．

Let (Xt)t>o be a standard Brownian motion in Rd, and let Br= {x: x Rd, [x] < r}, the ball centered at 0 with radius r. Denoting by Lr, rr and Tr the last exit time, the first exit time and the occupation time of Br, general formulas for the moments of above three random variables are given. Furthermore, the first and second moments are explicitluy determined.