If ${\displaystyle (W(t):t\geq 0)}$  is a Wiener process, and ${\displaystyle a>0}$  is a threshold (also called a crossing point), then the lemma states:

${\displaystyle \mathbb {P} \left(\sup _{0\leq s\leq t}W(s)\geq a\right)=2\mathbb {P} (W(t)\geq a)}$ 

Assuming ${\displaystyle W(0)=0}$  , due to the continuity of Wiener processes, each path (one sampled realization) of Wiener process on ${\displaystyle (0,t)}$  which finishes at or above value/level/threshold/crossing point ${\displaystyle a}$  the time ${\displaystyle t}$  ( ${\displaystyle W(t)\geq a}$  ) must have crossed (reached) a threshold ${\displaystyle a}$  ( ${\displaystyle W(t_{a})=a}$  ) at some earlier time ${\displaystyle t_{a}\leq t}$  for the first time . (It can cross level ${\displaystyle a}$  multiple times on the interval ${\displaystyle (0,t)}$ , we take the earliest.)

For every such path, you can define another path ${\displaystyle W'(t)}$  on ${\displaystyle (0,t)}$  that is reflected or vertically flipped on the sub-interval ${\displaystyle (t_{a},t)}$  symmetrically around level ${\displaystyle a}$  from the original path. These reflected paths are also samples of the Wiener process reaching value ${\displaystyle W'(t_{a})=a}$  on the interval ${\displaystyle (0,t)}$ , but finish below ${\displaystyle a}$ . Thus, of all the paths that reach ${\displaystyle a}$  on the interval ${\displaystyle (0,t)}$ , half will finish below ${\displaystyle a}$ , and half will finish above. Hence, the probability of finishing above ${\displaystyle a}$  is half that of reaching ${\displaystyle a}$ .

In a stronger form, the reflection principle says that if ${\displaystyle \tau }$  is a stopping time then the reflection of the Wiener process starting at ${\displaystyle \tau }$ , denoted ${\displaystyle (W^{\tau }(t):t\geq 0)}$ , is also a Wiener process, where:

${\displaystyle W^{\tau }(t)=W(t)\chi _{\left\{t\leq \tau \right\}}+(2W(\tau )-W(t))\chi _{\left\{t>\tau \right\}}}$ 

and the indicator function ${\displaystyle \chi _{\{t\leq \tau \}}={\begin{cases}1,&{\text{if }}t\leq \tau \\0,&{\text{otherwise }}\end{cases}}}$  and ${\displaystyle \chi _{\{t>\tau \}}}$ is defined similarly. The stronger form implies the original lemma by choosing ${\displaystyle \tau =\inf \left\{t\geq 0:W(t)=a\right\}}$ .

The earliest stopping time for reaching crossing point 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 ${\displaystyle \tau _{a}}$ , given by ${\displaystyle X_{t}:=W(t+\tau _{a})-a}$ , is also simple Brownian motion independent of ${\displaystyle {\mathcal {F}}_{\tau _{a}}^{W}}$ . Then the probability distribution for the last time ${\displaystyle W(s)}$  is at or above the threshold ${\displaystyle a}$  in the time interval ${\displaystyle [0,t]}$  can be decomposed as

${\displaystyle {\begin{aligned}\mathbb {P} \left(\sup _{0\leq s\leq t}W(s)\geq a\right)&=\mathbb {P} \left(\sup _{0\leq s\leq t}W(s)\geq a,W(t)\geq a\right)+\mathbb {P} \left(\sup _{0\leq s\leq t}W(s)\geq a,W(t)<a\right)\\&=\mathbb {P} \left(W(t)\geq a\right)+\mathbb {P} \left(\sup _{0\leq s\leq t}W(s)\geq a,X(t-\tau _{a})<0\right)\\\end{aligned}}}$ .

By the tower property for conditional expectations, the second term reduces to:

${\displaystyle {\begin{aligned}\mathbb {P} \left(\sup _{0\leq s\leq t}W(s)\geq a,X(t-\tau _{a})<0\right)&=\mathbb {E} \left[\mathbb {P} \left(\sup _{0\leq s\leq t}W(s)\geq a,X(t-\tau _{a})<0|{\mathcal {F}}_{\tau _{a}}^{W}\right)\right]\\&=\mathbb {E} \left[\chi _{\sup _{0\leq s\leq t}W(s)\geq a}\mathbb {P} \left(X(t-\tau _{a})<0|{\mathcal {F}}_{\tau _{a}}^{W}\right)\right]\\&={\frac {1}{2}}\mathbb {P} \left(\sup _{0\leq s\leq t}W(s)\geq a\right),\end{aligned}}}$ 

since ${\displaystyle X(t)}$  is a standard Brownian motion independent of ${\displaystyle {\mathcal {F}}_{\tau _{a}}^{W}}$  and has probability ${\displaystyle 1/2}$  of being less than ${\displaystyle 0}$ . The proof of the lemma is completed by substituting this into the second line of the first equation.^[2]

${\displaystyle {\begin{aligned}\mathbb {P} \left(\sup _{0\leq s\leq t}W(s)\geq a\right)&=\mathbb {P} \left(W(t)\geq a\right)+{\frac {1}{2}}\mathbb {P} \left(\sup _{0\leq s\leq t}W(s)\geq a\right)\\\mathbb {P} \left(\sup _{0\leq s\leq t}W(s)\geq a\right)&=2\mathbb {P} \left(W(t)\geq a\right)\end{aligned}}}$ .

The reflection principle is often used to simplify distributional properties of Brownian motion. Considering Brownian motion on the restricted interval ${\displaystyle (W(t):t\in [0,1])}$  then the reflection principle allows us to prove that the location of the maxima ${\displaystyle t_{\text{max}}}$ , satisfying ${\displaystyle W(t_{\text{max}})=\sup _{0\leq s\leq 1}W(s)}$ , has the arcsine distribution. This is one of the Lévy arcsine laws.^[3]