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 theorem 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 ( W ( t ) : t ≥ 0 ) {\displaystyle (W(t):t\geq 0)} is a Wiener process, and a > 0 {\displaystyle a>0} is a threshold (also called a crossing point), then the theorem states:
P ( sup 0 ≤ s ≤ t W ( s ) ≥ a ) = 2 P ( W ( t ) ≥ a ) {\displaystyle \mathbb {P} \left(\sup _{0\leq s\leq t}W(s)\geq a\right)=2\mathbb {P} (W(t)\geq a)}
Assuming W ( 0 ) = 0 {\displaystyle W(0)=0} , due to the continuity of Wiener processes, each path (one sampled realization) of Wiener process on ( 0 , t ) {\displaystyle (0,t)} which finishes at or above value/level/threshold/crossing point a {\displaystyle a} the time t {\displaystyle t} ( W ( t ) ≥ a {\displaystyle W(t)\geq a} ) must have crossed (reached) a threshold a {\displaystyle a} ( W ( t a ) = a {\displaystyle W(t_{a})=a} ) at some earlier time t a ≤ t {\displaystyle t_{a}\leq t} for the first time . (It can cross level a {\displaystyle a} multiple times on the interval ( 0 , t ) {\displaystyle (0,t)} , we take the earliest.) For every such path, you can define another path W ′ ( t ) {\displaystyle W'(t)} on ( 0 , t ) {\displaystyle (0,t)} that is reflected or vertically flipped on the sub-interval ( t a , t ) {\displaystyle (t_{a},t)} symmetrically around level a {\displaystyle a} from the original path. These reflected paths are also samples of the Wiener process reaching value W ′ ( t a ) = a {\displaystyle W'(t_{a})=a} on the interval ( 0 , t ) {\displaystyle (0,t)} , but finish below a {\displaystyle a} . Thus, of all the paths that reach a {\displaystyle a} on the interval ( 0 , t ) {\displaystyle (0,t)} , half will finish below a {\displaystyle a} , and half will finish above. Hence, the probability of finishing above a {\displaystyle a} is half that of reaching a {\displaystyle a} . In a stronger form, the reflection principle says that if τ {\displaystyle \tau } is a stopping time then the reflection of the Wiener process starting at τ {\displaystyle \tau } , denoted ( W τ ( t ) : t ≥ 0 ) {\displaystyle (W^{\tau }(t):t\geq 0)} , is also a Wiener process, where:
W τ ( t ) = W ( t ) χ { t ≤ τ } + ( 2 W ( τ ) − W ( t ) ) χ { t > τ } {\displaystyle W^{\tau }(t)=W(t)\chi _{\left\{t\leq \tau \right\}}+(2W(\tau )-W(t))\chi _{\left\{t>\tau \right\}}}
and the indicator function χ { t ≤ τ } = { 1 , if t ≤ τ 0 , otherwise {\displaystyle \chi _{\{t\leq \tau \}}={\begin{cases}1,&{\text{if }}t\leq \tau \\0,&{\text{otherwise }}\end{cases}}} and χ { t > τ } {\displaystyle \chi _{\{t>\tau \}}} is defined similarly. The stronger form implies the original theorem by choosing τ = inf { t ≥ 0 : W ( t ) = a } {\displaystyle \tau =\inf \left\{t\geq 0:W(t)=a\right\}} .
Proof The earliest stopping time for reaching crossing point a, τ a := inf { t : W ( t ) = a } {\displaystyle \tau _{a}:=\inf \left\{t:W(t)=a\right\}} , 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 {\displaystyle \tau _{a}} , given by X t := W ( t + τ a ) − a {\displaystyle X_{t}:=W(t+\tau _{a})-a} , is also simple Brownian motion independent of F τ a W {\displaystyle {\mathcal {F}}_{\tau _{a}}^{W}} . Then the probability distribution for the last time W ( s ) {\displaystyle W(s)} is at or above the threshold a {\displaystyle a} in the time interval [ 0 , t ] {\displaystyle [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 , W ( t ) − W ( τ a ) < 0 ) {\displaystyle {\begin{aligned}\mathbb {P} \left(\sup _{0\leq s\leq t}W(s)\geq a\right)&=\mathbb {P} \left(\sup _{0\leq s\leq t}W(s)\geq a,W(t)\geq a\right)+\mathbb {P} \left(\sup _{0\leq s\leq t}W(s)\geq a,W(t)<a\right)\\&=\mathbb {P} \left(W(t)\geq a\right)+\mathbb {P} \left(\sup _{0\leq s\leq t}W(s)\geq a,W(t)-W(\tau _{a})<0\right)\\\end{aligned}}} . By the strong markov property, W ( t ) − W ( τ a ) = D W ′ ( t − τ a ) {\displaystyle W(t)-W(\tau _{a}){\overset {\mathcal {D}}{=}}W'(t-\tau _{a})} where W ′ {\displaystyle W'} is a second simple brownian motion independent of { W ( u ) : 0 ≤ u ≤ τ a } {\displaystyle \{W(u):0\leq u\leq \tau _{a}\}} . Thus, by independence, the second term becomes:
P ( sup 0 ≤ s ≤ t W ( s ) ≥ a , W ( t ) − W ( τ a ) < 0 ) = P ( sup 0 ≤ s ≤ t W ( s ) ≥ a , W ′ ( t − τ a ) < 0 ) = P ( sup 0 ≤ s ≤ t W ( s ) ≥ a ) P ( W ′ ( t − τ a ) < 0 ) = 1 2 P ( sup 0 ≤ s ≤ t W ( s ) ≥ a ) , {\displaystyle {\begin{aligned}\mathbb {P} \left(\sup _{0\leq s\leq t}W(s)\geq a,W(t)-W(\tau _{a})<0\right)&=\mathbb {P} \left(\sup _{0\leq s\leq t}W(s)\geq a,W'(t-\tau _{a})<0\right)\\&=\mathbb {P} \left(\sup _{0\leq s\leq t}W(s)\geq a\right)\mathbb {P} \left(W'(t-\tau _{a})<0\right)\\&={\frac {1}{2}}\mathbb {P} \left(\sup _{0\leq s\leq t}W(s)\geq a\right),\end{aligned}}} . Since W ′ ( t ) {\displaystyle W'(t)} is a standard Brownian motion independent of F τ a W {\displaystyle {\mathcal {F}}_{\tau _{a}}^{W}} and has probability 1 / 2 {\displaystyle 1/2} of being less than 0 {\displaystyle 0} . The proof of the theorem 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 ) {\displaystyle {\begin{aligned}\mathbb {P} \left(\sup _{0\leq s\leq t}W(s)\geq a\right)&=\mathbb {P} \left(W(t)\geq a\right)+{\frac {1}{2}}\mathbb {P} \left(\sup _{0\leq s\leq t}W(s)\geq a\right)\\\mathbb {P} \left(\sup _{0\leq s\leq t}W(s)\geq a\right)&=2\mathbb {P} \left(W(t)\geq a\right)\end{aligned}}} .
Consequences 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 ] ) {\displaystyle (W(t):t\in [0,1])} then the reflection principle allows us to prove that the location of the maxima t max {\displaystyle t_{\text{max}}} , satisfying W ( t max ) = sup 0 ≤ s ≤ 1 W ( s ) {\displaystyle W(t_{\text{max}})=\sup _{0\leq s\leq 1}W(s)} , has the arcsine distribution. This is one of the Lévy arcsine laws.
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