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.

Simulation of Wiener process (black curve). When the process reaches the crossing point at a=50 at t$\approx$3000, both the original process and its reflection (red curve) about the a=50 line (blue line) are shown. After the crossing point, both black and red curves have the same distribution.

Statement

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

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

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

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

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

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

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

Proof

The earliest stopping time for reaching crossing point a, $\tau _{a}:=\inf \left\{t:W(t)=a\right\}$, is an almost surely bounded stopping time. Then we can apply the strong Markov property to deduce that a relative path subsequent to $\tau _{a}$, given by $X_{t}:=W(t+\tau _{a})-a$, is also simple Brownian motion independent of ${\mathcal {F}}_{\tau _{a}}^{W}$. 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

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

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

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

since $X(t)$ is a standard Brownian motion independent of ${\mathcal {F}}_{\tau _{a}}^{W}$ 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.

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

Consequences

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