brownian motion in stopping time

[Marina9]

Hello everyone!
I am stuck with a little problem concerning BM and stopping time.
If we define tau_n:=inf{t>=0 : |W_t|>= n}$, does anyone have idea how to show that |W(tau_n)|=n? I found one comment that on random interval [0,tau_n], Brownian motion is bounded by n, but it doesn't include any comment whatsoever.
Thank you very much!

 

[royhaas]

Since BM has continuous sample paths, and variance equal to \(\displaystyle t\) (in your formulation), it would seem that \(\displaystyle |W(\tau_n)| = n\), almost by definition. There may be a tacit assumption that \(\displaystyle n\) is an integer, but that doesn't appeaqr necessary because of path continuity.

 

[Marina9]

Continuity of sample paths is enough for this statement. Thank you very much for your help!