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.^[1] 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.

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

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

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

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

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

${\displaystyle W^{\tau }(t)=W(t)\chi _{\left\{t\leq \tau \right\}}+(2W(\tau )-W(t))\chi _{\left\{t>\tau \right\}}}${\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 ${\displaystyle \chi _{\{t\leq \tau \}}={\begin{cases}1,&{\text{if }}t\leq \tau \0,円&{\text{otherwise }}\end{cases}}}${\displaystyle \chi _{\{t\leq \tau \}}={\begin{cases}1,&{\text{if }}t\leq \tau \0,円&{\text{otherwise }}\end{cases}}} and ${\displaystyle \chi _{\{t>\tau \}}}${\displaystyle \chi _{\{t>\tau \}}} is defined similarly. The stronger form implies the original theorem by choosing ${\displaystyle \tau =\inf \left\{t\geq 0:W(t)=a\right\}}${\displaystyle \tau =\inf \left\{t\geq 0:W(t)=a\right\}}.

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

${\displaystyle {\begin{aligned}\mathbb {P} \left(\sup _{0\leq s\leq t}W(s)\geq a\right)&=\mathbb {P} \left(\sup _{0\leq s\leq t}W(s)\geq a,W(t)\geq a\right)+\mathbb {P} \left(\sup _{0\leq s\leq t}W(s)\geq a,W(t)<a\right)\\&=\mathbb {P} \left(W(t)\geq a\right)+\mathbb {P} \left(\sup _{0\leq s\leq t}W(s)\geq a,W(t)-W(\tau _{a})<0\right)\\\end{aligned}}}${\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, ${\displaystyle W(t)-W(\tau _{a}){\overset {\mathcal {D}}{=}}W'(t-\tau _{a})}${\displaystyle W(t)-W(\tau _{a}){\overset {\mathcal {D}}{=}}W'(t-\tau _{a})} where ${\displaystyle W'}${\displaystyle W'} is a second simple brownian motion independent of ${\displaystyle \{W(u):0\leq u\leq \tau _{a}\}}${\displaystyle \{W(u):0\leq u\leq \tau _{a}\}}. Thus, by independence, the second term becomes:

${\displaystyle {\begin{aligned}\mathbb {P} \left(\sup _{0\leq s\leq t}W(s)\geq a,W(t)-W(\tau _{a})<0\right)&=\mathbb {P} \left(\sup _{0\leq s\leq t}W(s)\geq a,W'(t-\tau _{a})<0\right)\\&=\mathbb {P} \left(\sup _{0\leq s\leq t}W(s)\geq a\right)\mathbb {P} \left(W'(t-\tau _{a})<0\right)\\&={\frac {1}{2}}\mathbb {P} \left(\sup _{0\leq s\leq t}W(s)\geq a\right),\end{aligned}}}${\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 ${\displaystyle W'(t)}${\displaystyle W'(t)} is a standard Brownian motion independent of ${\displaystyle {\mathcal {F}}_{\tau _{a}}^{W}}${\displaystyle {\mathcal {F}}_{\tau _{a}}^{W}} and has probability ${\displaystyle 1/2}${\displaystyle 1/2} of being less than ${\displaystyle 0}${\displaystyle 0}. The proof of the theorem is completed by substituting this into the second line of the first equation.^[2]

${\displaystyle {\begin{aligned}\mathbb {P} \left(\sup _{0\leq s\leq t}W(s)\geq a\right)&=\mathbb {P} \left(W(t)\geq a\right)+{\frac {1}{2}}\mathbb {P} \left(\sup _{0\leq s\leq t}W(s)\geq a\right)\\\mathbb {P} \left(\sup _{0\leq s\leq t}W(s)\geq a\right)&=2\mathbb {P} \left(W(t)\geq a\right)\end{aligned}}}${\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 ${\displaystyle (W(t):t\in [0,1])}${\displaystyle (W(t):t\in [0,1])} then the reflection principle allows us to prove that the location of the maxima ${\displaystyle t_{\text{max}}}${\displaystyle t_{\text{max}}}, satisfying ${\displaystyle W(t_{\text{max}})=\sup _{0\leq s\leq 1}W(s)}${\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.^[3]

• Stochastic calculus
• Theorems in probability theory

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