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.
If ( W ( t ) : t ≥ 0 ) is a Wiener process, and a > 0 is a threshold (also called a crossing point), then the lemma states:
P ( sup 0 ≤ s ≤ t W ( s ) ≥ a ) = 2 P ( W ( t ) ≥ a ) In a stronger form, the reflection principle says that if τ is a stopping time then the reflection of the Wiener process starting at τ , denoted ( W τ ( t ) : t ≥ 0 ) , is also a Wiener process, where:
W τ ( t ) = W ( t ) χ { t ≤ τ } + ( 2 W ( τ ) − W ( t ) ) χ { t > τ } and the indicator function χ { t ≤ τ } = { 1 , if t ≤ τ 0 , otherwise and χ { t > τ } is defined similarly. The stronger form implies the original lemma by choosing τ = inf { t ≥ 0 : W ( t ) = a } .
The earliest stopping time for reaching crossing point a, τ a := inf { t : W ( t ) = a } , is an almost surely bounded stopping time. Then we can apply the strong Markov property to deduce that a relative path subsequent to τ a , given by X t := W ( t + τ a ) − a , is also simple Brownian motion independent of F τ a W . Then the probability distribution for the last time W ( s ) is at or above the threshold a in the time interval [ 0 , t ] can be decomposed as
P ( sup 0 ≤ s ≤ t W ( s ) ≥ a ) = P ( sup 0 ≤ s ≤ t W ( s ) ≥ a , W ( t ) ≥ a ) + P ( sup 0 ≤ s ≤ t W ( s ) ≥ a , W ( t ) < a ) = P ( W ( t ) ≥ a ) + P ( sup 0 ≤ s ≤ t W ( s ) ≥ a , X ( t − τ a ) < 0 ) .
By the tower property for conditional expectations, the second term reduces to:
P ( sup 0 ≤ s ≤ t W ( s ) ≥ a , X ( t − τ a ) < 0 ) = E [ P ( sup 0 ≤ s ≤ t W ( s ) ≥ a , X ( t − τ a ) < 0 | F τ a W ) ] = E [ χ sup 0 ≤ s ≤ t W ( s ) ≥ a P ( X ( t − τ a ) < 0 | F τ a W ) ] = 1 2 P ( sup 0 ≤ s ≤ t W ( s ) ≥ a ) , since X ( t ) is a standard Brownian motion independent of F τ a W and has probability 1 / 2 of being less than 0 . The proof of the lemma is completed by substituting this into the second line of the first equation.
P ( sup 0 ≤ s ≤ t W ( s ) ≥ a ) = P ( W ( t ) ≥ a ) + 1 2 P ( sup 0 ≤ s ≤ t W ( s ) ≥ a ) P ( sup 0 ≤ s ≤ t W ( s ) ≥ a ) = 2 P ( W ( t ) ≥ a ) .
The reflection principle is often used to simplify distributional properties of Brownian motion. Considering Brownian motion on the restricted interval ( W ( t ) : t ∈ [ 0 , 1 ] ) then the reflection principle allows us to prove that the location of the maxima t max , satisfying W ( t max ) = sup 0 ≤ s ≤ 1 W ( s ) , has the arcsine distribution. This is one of the Lévy arcsine laws.