Reflection Principle for Brownian Motion

Medium

Exercise 2 / 4

Let $(W_t)_{t \geq 0}$ be a standard Brownian motion and define:

M_t := \max_{0 \leq s \leq t} W_s.

Question: Show that for any $a > 0$ :

\mathbb{P}(M_t \geq a) = 2\,\mathbb{P}(W_t \geq a) = \mathbb{P}(|W_t| \geq a).

a)=

P(∣Wt∣≥

a).