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Let $W_t$ be a Wiener process and for $a\geq0$

$$\tau_a:=\inf \left\{ t\geq0: |W_t|=\sqrt{at+7} \right\}.$$

Is $\tau_a<\infty$ almost everywhere? What about $E(\tau_a)$ then?

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    By the law of the iterated logarithm, you should have $\tau_a<\infty$.2012-02-17
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    So I could use ${\limsup_{t\rightarrow \infty} \frac{W_t}{\sqrt{2t\cdot ln(ln(t))}}=1}$ almost surely. But still it is $\sqrt{at+7} \geq \sqrt{2tln(ln(t))}$, for $a>2$, isn't it?2012-02-18
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    Got something out of the answer?2012-03-08

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