Reflection principle (Wiener process)

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Simulation of Wiener process (black curve). When the process reaches the crossing point at a=50 at t\approx3000, 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[edit]

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[edit]

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

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

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

Consequences[edit]

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[edit]

  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.