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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?

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  • $\begingroup$ 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. $\endgroup$ Commented Dec 25, 2017 at 12:31
  • $\begingroup$ Thank you for your advice and I hope to see some other beautiful resolutions to this question. $\endgroup$
    – Ghost W
    Commented Dec 26, 2017 at 6:55

1 Answer 1

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Now I know a method by applying martingale optional stopping theorem.

Proof:

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 $Var \tau=2/3$. $\qquad \blacksquare$

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