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Lec28 - Stat 150 Stochastic Processes Spring 2009 Lecture...

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Stat 150 Stochastic Processes Spring 2009 Lecture 28: Brownian Bridge Lecturer: Jim Pitman From last time: Problem : Find hitting probability for BM with drift δ , D t := a + δt + B t , P ( δ ) a ( D t hits b before 0). Idea : Find a suitable MG. Found e - 2 δD t is a MG. Use this : Under P a start at a . E a e - 2 δD 0 = e - 2 δa . Let T = first time D t hits 0 or b . Stop the MG at time T , M t = e - 2 δD t . Look at ( M t T , t 0). For simplicity, take δ > 0, then 1 e - 2 δD t > e - 2 δa > e - 2 δb > 0 . For 0 t T , get a MG with values in [0,1] when we stop at T . Final value of MG = 1 · 1 (hit 0 before b ) + e - 2 δb · 1 (hit b before 0) 1 = P (hit b ) + P (hit 0) e - 2 δa = 1 · P (hit 0) + e - 2 δb · P (hit b ) = P (hit b ) = 1 - e - 2 δa 1 - e - 2 δb Equivalently, start at 0, P ( B t + δt ever reaches - a ) = e - 2 δa . Let M := - min t ( B t + δt ), then P ( M a ) = e - 2 δa = M Exp(2 δ ). Note the memoryless property : P ( M > a + b ) = P ( M > a ) P ( M > b ) This can be understood in terms of the strong Markov property of the drifting BM at its first hitting time of - a : to get below - ( a + b ) the process must get down to - a , and the process thereafter must get down a further amount - b . 0 a b 1
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Lecture 28: Brownian Bridge 2 Brownian Bridge This is a process obtained by conditioning on the value of B at a fixed time, say time=1. Look at ( B t , 0 t 1 | B 1 ), where B 0 = 0.
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