Note: This is a crosspost from http://math.stackexchange.com; the original question may be found here.
I have a question regarding a script by Greg Lawler on Bessel processes:
http://www.math.uchicago.edu/~lawler/bessel.pdf
There I encounter difficulties in understanding the last sentence on page 2.
Let $ W_t = (W_t^1,\dots,W_t^d) $ be a (standard) d-dimensional Brownian motion and
$$ X_t = |W_t| = |W_t|_2 = \left(\sum\limits_{j=1}^d \left(W_t^j\right)^2\right)^{1/2} $$
its (Euclidean) norm.
Now it is noted:
$$ dX_t^2=\sum_{j=1}^d d[(W_t^j)^2] = 2\sum_{j=1}^d W_t^j dW_t^j + d \;dt $$
and we are supposed to be allowed to write the later as
$$ dX_t^2 = d \; dt + 2X_t dZ_t $$
with
$$ Z_t = \sum_{j=1}^d\int_0^t\frac{|W_s^j|}{X_s} dW_s^j .$$
It is not clear to me why we may rewrite it like that. I think
$$ X_t dZ_t = \sum_{j=1}^d |W_t^j| dW_t^j $$ holds true.
Wouldn't this imply, e.g.,
$$ |W_t^1| dW_t^1 = W_t^1 dW_t^1? $$
But this is not true, is it? Do I have a lapse of thought here?
Thank you for any hints :)
Edit: So on SE someone suggested the $|\cdot|$ in the definition of $Z_t$ to be a typo. That is also what I was thinking lately.