Reflection Principle

Mirror a path after it first touches a level: the running maximum's tail is twice the endpoint's.

The idea

Theorem (Reflection principle).

Let $W$ be a standard Brownian motion and let $M_{T} = \max_{0 \le t \le T} W(t)$ be its running maximum. Then for every level $a > 0$, $\mathbb{P}(M_{T} \ge a) = 2\,\mathbb{P}(W(T) \ge a).$

The running maximum is the highest level the path reaches by time $T$. At a fixed time $t$ the position $W(t)$ is normal with variance $t$, which settles any question about one instant. The running maximum depends on the whole path, and the moment the path first reaches a level is itself random, so no single snapshot settles it. The theorem reduces it to one anyway: the right side is a single normal probability, so the law of the running maximum — and with it the law of the first time the path reaches a level — follows from the law of a single position.

Ways to work on it

• Walkthrough. Mirror a Brownian path where it first hits a level to find the law of its running maximum and its hitting times.
• Proof. Prove the reflection principle by pairing each path with its mirror image, and deduce the law of the running maximum.
• Practice. Running-maximum and hitting-time probabilities from the standard normal table, forwards and backwards.
• Hardest. Joint probabilities for a Brownian path's maximum and endpoint, and how a hitting time scales with the level.

Not sure where to start? Take the ten-question placement test.