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. 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:
 * $$ \mathbb{P} \left(\sup_{0 \leq s \leq t} W(s) \geq a \right) = 2\mathbb{P}(W(t) \geq a) $$

Assuming $$ W(0) = 0 $$, due to the continuity of Wiener processes, each path (one sampled realization) of Wiener process on $$ (0,t) $$ which finishes at or above value/level/threshold/crossing point $$a$$ the time $$ t $$ ( $$W(t) \geq a$$ ) must have crossed (reached) a threshold $$a$$ ( $$ W(t_a) = a $$ ) at some earlier time $$ t_a \leq t $$ 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 path $$W'(t)$$ on $$ (0,t) $$ that is reflected or vertically flipped on the sub-interval $$ (t_a,t) $$ symmetrically around level $$a$$ from the original path. These reflected paths are also samples of the Wiener process reaching value $$ W'(t_a) = a $$ on the interval $$(0,t)$$, but finish below $$a$$. Thus, of all the paths that reach $$a$$ on the interval $$ (0,t) $$, half will finish below $$a$$, and half will finish above. Hence, the probability of finishing above $$a$$ is half that of reaching $$a$$. 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:
 * $$ W^\tau(t) = W(t)\chi_\left\{t \leq \tau\right\} + (2W(\tau) - W(t))\chi_\left\{t > \tau\right\}$$

and the indicator function $$\chi_{\{t \leq \tau\}}= \begin{cases} 1, & \text{if }t \leq \tau \\ 0, & \text{otherwise }\end{cases}$$ and $$\chi_{\{t > \tau\}} $$is defined similarly. 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.



\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.