L11 - Lecture 11 Some Properties of Brownian Motion 11.1...

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Unformatted text preview: Lecture 11 Some Properties of Brownian Motion 11.1 Reflection principle and the first passage time Define M t = max u t W u and recall the first passage time ( a ) = inf { t 0 : W t = a } . Proposition 11.1 For every a > , P ( M t a ) = P ( ( a ) < t ) = 2 P ( W t > a ) = 2- 2 ( a/ t ) , (11.1) where ( ) denotes the cdf of standard normal distribution. Note: The rigorous proof relies on the strong Markov property of W , but drawing a picture would provide a heuristic argument. Furthermore, Theorem 10.1 implies that W * is also a Brownian motion with W * t = W ( a )+ t- W ( a ) . 11.2 Behaviors of sample paths Although being continuous, Brownian paths have crazy behaviors. 11.2.1 Zero set Proposition 11.2 Let S = { t 0 : W t = 0 } . Then with probability one, ( i ) meas ( S ) = 0 , where meas denotes the Lebesgue measure; ( ii ) in every interval ( t 1 ,t 2 ) IR + , there are infinitely many elements of S ; ( iii ) S is a perfect set, i.e.is a perfect set, i....
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This note was uploaded on 11/17/2011 for the course STOR 890 taught by Professor Staff during the Spring '08 term at UNC.

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L11 - Lecture 11 Some Properties of Brownian Motion 11.1...

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