Let W(t) be a standard Brownian motion Let a 0 Define Ta as the first time that W(t) a That is a Show that for any t 0, we have b Using Part (a), show that c Using Part (b), show that the PDF of T a is given by By symmetry of Brownian motion, we conclude that for any a 0, we have Ta min t W(t) a ...

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Let W(t) be a standard Brownian motion. Let a > 0. Define Ta as the first time

Question:

Let W(t) be a standard Brownian motion. Let a > 0. Define Ta as the first time that W(t) = a. That is

a. Show that for any t ≥ 0, we have

b. Using Part (a), show that P(Ta t)=21-0 a (+7)] t

c. Using Part (b), show that the PDF of T_a is given by fr. (t) = a t/2t p{-} exp 2t

By symmetry of Brownian motion, we conclude that for any a ≠ 0, we have fra (t) = lal t2t exp 2t

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Related Book For answer-question

Introduction To Probability Statistics And Random Processes

ISBN: 9780990637202

1st Edition

Authors: Hossein Pishro-Nik

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Question Posted: Nov 01, 2023 12:58 PM

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Let W(t) be a standard Brownian motion. Let a > 0. Define Ta as the first time

Question:

Ta = min{t: W(t) = a}.

Step by Step Answer:

Introduction To Probability Statistics And Random Processes

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