有界变差。
 \int_{0}^{t} B_{s} ds 是一个Riemann-Stieltjes integral,  这个东西，with probability 1 是有界变差的 因为 布朗运动 是
as 连续的。

thanks for your replies, but could you pls make it more clear by telling me how to define the Riemann-Stieltjes integral with respect to a Brownian Motion? it seems that the paths of BM are too rough to define Riemann-Stieltjes integral, althrough it is as continuous? Or is that all the continuous function can be defined a Riemann-Stieltjes integral ?

thanks in advance!

我好像觉得没有问题啊。
对于每一个 omega，我们都可以定义 这个积分。这样，almost surly，我们可以定义这个积分，而且通过
product rule，我们知道 \int_{0}^{t} B_{s} ds = B_{t}*t-\int_{0}^{t} sdB_{s}。

另外一个看法，
 \int_{0}^{t} B_{s} ds =\int_{0}^{t} B_{s}^(+) ds - \int_{0}^{t} B_{s}^{-} ds
可以写成两个 增函数之差。

http://en.wikipedia.org/wiki/Bounded_variation

Definition 2. A real-valued function f on the real line is said to be of bounded variation (BV function) on a chosen interval [a,b] if its total variation is finite

V=int_a^b{abs(Bs)}ds is finite, since B is continuous.