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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.

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Hint: For every real number $x$, $$\mathbf 1_{\{x \geqslant 0\}} + \mathbf 1_{\{x < 0\}}=1,\qquad x\mathbf 1_{\{x \geqslant 0\}}- x\mathbf 1_{\{x < 0\}}= |x| $$

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So this confirms the two identifies path-wise. (i.e. for $W_t(\omega)$ and $|W_t(\omega)|$). However, I'm not quite sure how this can then be extended to demonstrate the case for the full $W_t$ random variable? – Jase Nov 8 '12 at 17:08
What you call the case for the full random variable is not clear to me. Random variables $X,Y:\Omega\to\mathbb R$ are such that $X=Y$ (almost surely) if and only if $X(\omega)=Y(\omega)$ for (almost) every $\omega$ in $\Omega$. The identities in my post show this is the case here. – Did Nov 8 '12 at 17:23
Okay, so you're suggesting that I use the fact that the identities I've stated hold path-wise (i.e. $\text{P-a.s.}$ under filtration equipped probability space with measure $P$), and therefore hold in all possible states of the world, and therefore the expressions are correct as they're stated. – Jase Nov 8 '12 at 17:27
One can forget the filtration context altogether. There are only functions, defined on some set $\Omega$. – Did Nov 8 '12 at 17:29
Your previous-to-last comment is correct (and one can omit almost surely in it). – Did Nov 9 '12 at 13:32

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