Reflection principle (Wiener process)
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. Statement
If  is a Wiener process, and  is a threshold (also called a crossing point), then the lemma states:

Assuming  , due to the continuity of Wiener processes, each path (one sampled realization) of Wiener process on  which finishes at or above value/level/threshold/crossing point  the time   (  ) must have crossed (reached) a threshold  (  ) at some earlier time  for the first time . (It can cross level  multiple times on the interval , we take the earliest.) For every such path, you can define another path  on  that is reflected or vertically flipped on the sub-interval  symmetrically around level  from the original path. These reflected paths are also samples of the Wiener process reaching value  on the interval , but finish below . Thus, of all the paths that reach  on the interval , half will finish below , and half will finish above. Hence, the probability of finishing above  is half that of reaching . In a stronger form, the reflection principle says that if  is a stopping time then the reflection of the Wiener process starting at , denoted , is also a  Wiener process, where:

and the indicator function  and is defined similarly. The stronger form implies the original lemma by choosing . Proof
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
. By the tower property for conditional expectations, the second term reduces to:

since  is a standard Brownian motion independent of  and has probability  of being less than . The proof of the lemma is completed by substituting this into the second line of the first equation. . Consequences
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. References

Stochastic calculus
Probability theorems