Setup
Let
$$ T=\inf\{t\ge 0:B_t=-a \text{ or } B_t=b\}, \qquad a,b>0. $$We want
$$ \mathbb P(B_T=b). $$Martingale used
Use the martingale $(B_t)$.
Before time $T$, the path stays in $[-a,b]$, so the stopped process is bounded and OST applies:
$$ \mathbb E[B_T]=\mathbb E[B_0]=0. $$Solve for the probability
Since $B_T\in\{-a,b\}$, letting
$$ p=\mathbb P(B_T=b), $$we have
$$ bp-a(1-p)=0. $$Hence
$$ p=\frac{a}{a+b}. $$Source Links
- STA447
- Phase 6 Atomic Reading Order
- Phase 6 — Brownian Motion 基础
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