Reflection principle (Wiener process)

From Wikipedia, the free encyclopedia

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.[1] 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.

Wiener_process_and_its_reflection_upon_reaching_a_crossing_point.png
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.