Show $ \int _0^t \frac{\left|B_u \right|}{u}du < \infty \ a.e.$

Question

How to show that for all $t\geq 0$ $$ \int _0^t \frac{\left|B_u \right|}{u}du < \infty \ a.e.,$$ where $ \left( B_t \right)_{t\geq 0}$ is the real standard brownian motion starting from zero ?

Answer 1

Using Fubini-Tonelli theorem and Cauchy-Schwarz's inequality, we have $$ E\left[\int_0^t \frac{|B_u|}{u}\,du\right] = \int_0^t\frac{E(|B_u|)}{u}\,du \leq \int_0^t \frac{E(B_u^2)^{1/2}}{u}\,du = \int_0^t \frac{du}{\sqrt{u}} =2\sqrt{t} <\infty. $$

Edit: As noted by did, the expectation can be computed exactly:

$$ E\left[\int_0^t \frac{|B_u|}{u}\,du\right] = \sqrt{\frac{8t}{\pi}}. $$

Answer 2

Hint: Show that $$ E\left[\int_0^t \frac{|B_u|}{u}\,\mathrm du\right]<\infty $$ by the use of Tonelli's theorem.