Setup

Let

$$ T_a=\inf\{t>0:B_t\ge a\}, \qquad a>0. $$

Then

$$ {T_a\le t}

\left{\max_{0\le s\le t} B_s\ge a\right}. $$

Main identity

The reflection principle gives

$$ \mathbb P(T_a\le t)=2\mathbb P(B_t\ge a). $$

Equivalently,

$$ \mathbb P(T_a\le t)=2\Phi\left(-\frac{a}{\sqrt t}\right). $$

Why it works

Once Brownian motion first hits the level $a$, the path after that time can be reflected. By symmetry, paths ending above $a$ correspond to paths ending below $a$ after reflection.

Source Links

• STA447
• Phase 6 Atomic Reading Order
• Phase 6 — Brownian Motion 基础
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