Simulation of Wiener process (black curve). When the process reaches the crossing point at a=50 at t3000, both the original process and its reflection (red curve) about the a=50 line (blue line) are shown. After the crossing point, both black and red curves have the same distribution.
In the theory of probability for stochastic processes, the reflection principle for a Wiener process states that if the path of a Wiener process f(t) reaches a value f(s) = a at time t = s, then the subsequent path after time s has the same distribution as the reflection of the subsequent path about the value a. More formally, the reflection principle refers to a lemma concerning the distribution of the supremum of the Wiener process, or Brownian motion. The result relates the distribution of the supremum of Brownian motion up to time t to the distribution of the process at time t. It is a corollary of the strong Markov property of Brownian motion.
The earliest stopping time for reaching crossing point a, , is an almost surely bounded stopping time. Then we can apply the strong Markov property to deduce that a relative path subsequent to , given by , is also simple Brownian motion independent of . Then the probability distribution for the last time is at or above the threshold in the time interval can be decomposed as
The reflection principle is often used to simplify distributional properties of Brownian motion. Considering Brownian motion on the restricted interval then the reflection principle allows us to prove that the location of the maxima , satisfying , has the arcsine distribution. This is one of the Lévy arcsine laws.