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

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Lecture 11 Some Properties of Brownian Motion 11.1 Reflection principle and the first passage time Define M t = max 0 u t W u and recall the first passage time τ ( a ) = inf { t 0 : W t = a } . Proposition 11.1 For every a > 0 , P ( M t a ) = P ( τ ( a ) < t ) = 2 P ( W t > a ) = 2 - 2 Φ( a/ t ) , (11.1) where Φ( · ) 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 τ ( a )+ t - W τ ( 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. S is closed and dense in itself. Note: To show (i), it follows from Fubini’s Theorem that E [meas( S )] = E Z 0 I { W t =0 } dt = Z 0 P ( W t = 0) dt = 0 .

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

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