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 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${\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.

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 continuity of Wiener process, each path (one sampled realization) of Wiener process on (0,t) which ends up at or above value/level/threshold/crossing point 'a' the time t (${\displaystyle W(t)\geq a}$) must have crossed a threshold 'a' (${\displaystyle W(t_{a})=a)}$) at some earlier time ${\displaystyle t_{a}\leq t}$ 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 sampled path of Wiener process W' on (0,t) that is reflected or vertically flipped on the sub-interval ${\displaystyle (t_{a},t)}$ symmetrically around level 'a' from the original path.( ${\displaystyle a-W'(t)=W(t)-a}$ ) That reflected path also reached value ${\displaystyle W'(t_{a})=a}$ on the interval (0,t) and is also a Wiener process or Brownian motion. Both original and reflected paths form set of paths that reach value 'a' on (0,t) and they are twice as many as those that end up at or above threshold 'a' (original path only) at time t. If each path is equally probable (imagine symmetric random walk from 0 on trees) then reaching threshold 'a' at any time on (0,t) is twice as likely as ending up at or above threshold 'a' at the time t. What about paths that reach level 'a' on (0,t) and then end up somewhere at value ${\displaystyle W(t)<a}$ at the time t? Are they accounted for? Yes. There are exactly those reflected paths that are counted towards the number of the paths that reached threshold 'a' only and they are exactly as many as those that ended up above threshold 'a' at the time t. Once Wiener process reached threshold 'a' , then due to the symmetry there is an equal probability (p=0.5) it will end up above or below threshold 'a' at any future time t. So conditional probability: ${\displaystyle P(W(t)\geq a|W(t_{a})=a)=0.5}$. Paths with ${\displaystyle W(t)<a}$ that never reach threshold 'a' are never considered.

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.