Define $\tau_a = \inf \left\lbrace t \geq 0 | B(t) \geq a \right\rbrace $ for some $a>0$. The problem is to show that $ \tau_a \stackrel{d}{=} \sqrt a\tau_1 $. What I've done so far:
$$P(\tau_a \leq t) = 2P(B(t) \geq a) = 2P \left(Z \geq \frac{a}{\sqrt t}\right)$$
The first step is from the reflection principle, the second is just normalizing the Brownian Motion. But now doing something similar:
$$P(\sqrt a\tau_1 \leq t) = 2P\left(B \left(\frac{t}{\sqrt a} \right) \geq 1 \right) = 2P \left(Z \geq \left(\frac{\sqrt a}{t}\right)^{\frac{1}{2}} \right)$$
Clearly these are different so I'm not sure where the mistake I'm making is coming in. I've already calculated a density function for $\tau_a$ and showed that $\tau_a$ has stationary, independent increments, but I can't see where that fact would be helpful, if anywhere.
Best Answer
Your calculations are correct, but the claim is not. Instead of
it should read
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