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Let $B_t$ be a 2D Brownian Motion with $B_0 = (1,0)$. Now, express $B_t$ in polars, that is, $B_t = (r(t), \theta(t))$. Let $\tau = \inf\{t > 0 : \theta(t) \geq 2 \pi \}$. What is $\mathbb{P}[\tau \leq 1]$?

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  • $\begingroup$ This is an interesting question, what have you tried to far? $\endgroup$ – nullUser Oct 22 '13 at 22:23
  • $\begingroup$ @nullUser I was trying to model the problem with a differential equation. For example, the probability distribution of $B_t$ satisfies the heat equation and one can certainly solve this equation in polars and impose appropriate initial and boundary conditions. However, I can't link this probability distribution and the distribution of $\tau$. I have also tried to derive the distribution of $\theta$ or $\tau$ with martingale theory, but have not succeeded. $\endgroup$ – rodms Oct 22 '13 at 22:31

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