Consider a Wiener process $W_t$ which is adapted to $\mathscr{F}_t$, where this filtration has all of the standard properties. I'm also working with a stock-standard probability space here.
I want to know if the following useful identities are correct:
$W_t = {1}_{\{W_t \geq 0\}}W_t + {1}_{\{W_t < 0\}}W_t$
$|W_t| = {1}_{\{W_t \geq 0\}}W_t - {1}_{\{W_t < 0\}}W_t$
Note that I mean "$=$" as actually equal and not only equal in distribution.
Hint: For every real number $x$, $$\mathbf 1_{\{x \geqslant 0\}} + \mathbf 1_{\{x < 0\}}=1,\qquad x\mathbf 1_{\{x \geqslant 0\}}- x\mathbf 1_{\{x < 0\}}= |x| $$