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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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    Your previous-to-last comment is correct (and one can omit *almost surely* in it).2012-11-09