# Variance of hitting time for Reflected Brownian Motion

Let $B_t$ to be a standard Brownian Motion and $Z_t=|B_t|$ a RBM.

Denote $\tau = \inf \{t>0: Z_t=1\}$, how to calculate $Var\ \tau \$ without/with martingale theory?

• Welcome to MSE. It will be more likely that you will get an answer if you show us that you made an effort.Welcome to MSE. – José Carlos Santos Dec 25 '17 at 12:31
• Thank you for your advice and I hope to see some other beautiful resolutions to this question. – Ghost W Dec 26 '17 at 6:55

Theorem : If $M_t$ is a martingale, $\tau$ is a stopping time, then we have $EM_\tau=EM_0$.
When $B_t$ is a standard Brownian Motion, we can verify that $M_t=B_t^2-t$ is a martingale. Hence by applying the theorem before, we hold that $E_\tau=1$.
Furthermore, $X_t=B_t^4-6tB^2_t+3t^2$ is also a martingale with mean $EX_t=0$. Therefore $E\tau^2=5/3$.
Combine this two equalities together and we have $Var \tau=2/3$. $\qquad \blacksquare$