The distribution of a standard Brownian motion X at a positive time t is, by definition, centered normal with variance t. What can we say about its maximum value up until the time? This is X^{∗}_{t} = sup_{s ≤ t}X_{s}, and is clearly nonnegative and at least as big as X_{t}. To be more precise, consider the probability that the maximum is greater than a fixed positive value a. Such problems will be familiar to anyone who has looked at pricing of financial derivatives such as barrier options, where the payoff of a trade depends on whether the maximum or minimum of an asset price has crossed a specified barrier level.

This can be computed with the aid of a symmetry argument commonly referred to as the reflection principle. The idea is that, if we reflect the Brownian motion when it first hits a level, then the resulting process is also a Brownian motion. The first time at which X hits level a is τ = inf{t ≥ 0: X_{t} ≥ a}, which is a stopping time. Reflecting the process about this level at all times after τ gives a new process