Reflection principle (Wiener process)

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.^[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 $(W(t): t \geq 0)$ 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 $\mathcal{F}^W_{\tau_a}$. 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

$\begin{align} \mathbb{P}\left(\sup_{0\leq s\leq t}W(s) \geq a\right) & = \mathbb{P}\left(\sup_{0\leq s\leq t}W(s) \geq a, W(t) \geq a\right) + \mathbb{P}\left(\sup_{0\leq s\leq t}W(s) \geq a, W(t) < a\right)\\ & = \mathbb{P}\left(W(t) \geq a\right) + \mathbb{P}\left(\sup_{0\leq s\leq t}W(s) \geq a, X(t-\tau_a) < 0\right)\\ \end{align}$.

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

$\begin{align} \mathbb{P}\left(\sup_{0\leq s\leq t}W(s) \geq a, X(t-\tau_a) < 0\right) &= \mathbb{E}\left[\mathbb{P}\left(\sup_{0\leq s\leq t}W(s) \geq a, X(t-\tau_a) < 0| \mathcal{F}^W_{\tau_a}\right)\right]\\ & = \mathbb{E}\left[\chi_{\sup_{0\leq s\leq t}W(s) \geq a} \mathbb{P}\left(X(t-\tau_a) < 0| \mathcal{F}^W_{\tau_a}\right)\right]\\ & = \frac{1}{2}\mathbb{P}\left(\sup_{0\leq s\leq t}W(s) \geq a\right) , \end{align}$

since $X(t)$ is a standard Brownian motion independent of $\mathcal{F}^W_{\tau_a}$ 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.^[2]

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

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.^[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.