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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 ${\displaystyle (W(t):t\geq 0)}$ 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 ${\displaystyle }$ 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:

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

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.

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

Consequences

The reflection principle is often used to simplify distributional properties of Brownian motion. Considering Brownian motion on the restricted interval ${\displaystyle (W(t):t\in )}$ 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.

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.

• Stochastic calculus
• Probability theorems