Basil Hiley

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.

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.

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:

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

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:

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

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

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

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

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

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 first equation.

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.

References

43 year old Petroleum Engineer Harry from Deep River, usually spends time with hobbies and interests like renting movies, property developers in singapore new condominium and vehicle racing. Constantly enjoys going to destinations like Camino Real de Tierra Adentro.