Reflection principle (Wiener process)

Simulation of Wiener process (black curve). When the process reaches the crossing point at a=50 at t${\displaystyle \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. ^[1] 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 ${\displaystyle (W(t):t\geq 0)}$ 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 ${\displaystyle \tau =\inf \left\{t\geq 0:W(t)=a\right\}}$.

Proof

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]

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

Consequences

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]

References

1. Jacobs, Kurt (2010). Stochastic Processes for Physicists. Cambridge University Press. pp. 57–59. ISBN   9781139486798.
2. Mörters, P.; Peres,Y. (2010) Brownian Motion, CUP. ISBN   978-0-521-76018-8
3. Lévy, Paul (1940). "Sur certains processus stochastiques homogènes". Compositio Mathematica. 7: 283–339. Retrieved 15 February 2013.