Proof of the Reflection Principle

We prove that reflecting Brownian motion about a level after hitting it produces another Brownian motion.

Proof

Let $\tau_a = \inf\{t : B_t = a\}$ and define

$\tilde{B}_t = \begin{cases} B_t & t \leq \tau_a, \\ 2a - B_t & t > \tau_a. \end{cases}$

Step 1: By the strong Markov property, $(B_{\tau_a+t} - a)_{t \geq 0}$ is a Brownian motion starting at 0, independent of $\mathcal{F}_{\tau_a}$.

Step 2: By symmetry of Brownian motion ($B_t \overset{d}{=} -B_t$), the process $(a - B_{\tau_a+t})$ is also a Brownian motion starting at $a$, independent of $\mathcal{F}_{\tau_a}$.

Step 3: Hence $\tilde{B}_t = 2a - B_t$ for $t > \tau_a$ has the same distribution as $B_t$.

Step 4: The reflected path $\tilde{B}$ has continuous trajectories and the correct finite-dimensional distributions, so $\tilde{B} \overset{d}{=} B$.

Application

The reflection principle immediately yields:

$\mathbb{P}(M_t \geq a) = \mathbb{P}(\tau_a \leq t) = 2\mathbb{P}(B_t \geq a),$

giving the distribution of the maximum. This formula is fundamental in option pricing for barrier derivatives.