Expressing a Wiener process with indicator functions.

[operationsres]

Homework Statement

Suppose that $W_t$ is a Wiener process (i.e. standard Brownian Motion).

Is it true that $W_t = \mathbf{1}_{\{W_t \geq 0\}}W_t + \mathbf{1}_{\{W_t < 0\}}W_t$ ?

The Attempt at a Solution

No idea. This isn't even homework it's part of my attempt of a solution to a homework problem. It feels correct, as the LHS = RHS in every possible state of the world. But stochastics has surprised me before!

We can at least show that they're equal in distribution, by checking if some of the conditions for a Wiener process hold:

$\text{var}(\mathbf{1}_{\{W_t \geq 0\}}W_t + \mathbf{1}_{\{W_t < 0\}}W_t) = (\mathbf{1}_{\{W_t \geq 0\}}W_t)^2 + (\mathbf{1}_{\{W_t < 0\}}W_t)^2$
$=\mathbf{1}_{\{W_t \geq 0\}}t + \mathbf{1}_{\{W_t < 0\}}t$
$= t$

That's good. Also;

$E[\mathbf{1}_{\{W_t \geq 0\}}W_t + \mathbf{1}_{\{W_t < 0\}}W_t | \mathscr{F}_0]$
$= 0$

Which is also good. Then the next steps would be to show that it's continuous everywhere and that increments are stationary and independent.

 

[kurt.physics]

But I don't know how to do that. This is a problem where I just have to guess and check, right? Any help would be greatly appreciated!