Brownian Motion Exit From an Interval
Let $B_t$ be a standard Brownian motion starting at $0$. Fix $a, b > 0$ and define the first exit time
$\tau = \inf\{t \geq 0 : B_t \notin (-b,\, a)\}$
so $\tau$ is the first time the process leaves the interval $(-b, a)$.
1. Compute $P(B_\tau = a)$ -- the probability that the process exits through the upper barrier.
2. Compute $E[\tau]$ -- the expected time to exit.
3. Interpret part (1) as the fair odds for a one-touch wager with asymmetric barriers. If a bookmaker offers a bet that pays $\
$ when the process first hits $a$ (and nothing if it hits $-b$ first), what is the fair price?
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