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Brownian meander

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Title: Brownian meander  
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Subject: Brownian excursion, Stochastic processes, Continuous-time stochastic process, Markov additive process, Poisson point process
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Brownian meander

In the mathematical theory of probability, Brownian meander W^+ = \{ W_t^+, t \in [0,1] \} is a continuous non-homogeneous Markov process defined as follows:

Let W = \{ W_t, t \geq 0 \} be a standard one-dimensional Brownian motion, and \tau := \sup \{ t \in [0,1] : W_t = 0 \} , i.e the last time before t = 1 when W visits \{ 0 \}. Then

W^+_t := \frac{1}{\sqrt{1 - \tau}} | W_{\tau + t (1-\tau)} |, \quad t \in [0,1].

The transition density p(s,x,t,y) \, dy := P(W^+_t \in dy \mid W^+_s = x) of Brownian meander is described as follows:

For 0 < s < t \leq 1 and x, y > 0, and writing

\phi_t(x):= \frac{\exp \{ -x^2/(2t) \}}{\sqrt{2 \pi t}} \quad \text{and} \quad \Phi_t(x,y):= \int^y_x\phi_t(w) \, dw,

we have

\begin{align} p(s,x,t,y) \, dy &:= P(W^+_t \in dy \mid W^+_s = x) \\ &= \bigl( \phi_{t-s}(y-x) - \phi_{t-s}(y+x) \bigl) \frac{\Phi_{1-t}(0,y)}{\Phi_{1-s}(0,x)} \, dy \end{align}


p(0,0,t,y) \, dy := P(W^+_t \in dy ) = 2\sqrt{2 \pi} \frac{y}{t}\phi_t(y)\Phi_{1-t}(0,y) \, dy.

In particular,

P(W^+_1 \in dy ) = y \exp \{ -y^2/2 \} \, dy, \quad y > 0,

i.e W^+_1 has the Rayleigh distribution with parameter 1, the same distribution as \sqrt{2 \mathbf{e}}, where \mathbf{e} is an exponential random variable with parameter 1.


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