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MathReview8

# MathReview8 - Exam Midterm Page 1of 3 Date University of...

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Unformatted text preview: Exam: Midterm Page: 1of 3 Date: Mar 12, 2009 University of Texas at Austin, Department of Mathematics M385D/CAM384K - Theory of Probability II The Midterm Exam ( Note: In all problems, ( B t ) t ∈ [0 , ∞ ) is a Brownian motion all of whose trajectories are continuous functions. The filtration ( F t ) t ∈ [0 , ∞ ) is given by F t = σ ( B s ; s ≤ t ). ) Problem 1.1 (30pts) . (1) (15pts) Give a precise definition of the Wiener measure (make sure to include the domain and explain how the σ-algebra on which it acts is generated). (2) (15pts) State the strong Markov property for the Brownian motion (make sure to include the description of the underlying filtration). Solution: See Notes. Problem 1.2 (35pts) . Define the process ( X t ) t ∈ [0 , ∞ ) by X t = R t B u du , for t ∈ [0 , ∞ ). (1) (20pts) Show that X t ∈ L 2 , for all t ∈ [0 , ∞ ), and, for s,t ∈ [0 , ∞ ), compute the covariance Cov [ X t ,X s ]. ( Note: You may use the fact that M t = sup s ≤ t B s has the same distribution as | B t | .) (2) (15pts) Is ( X t ) t ∈ [0 , ∞ ) an ( F t ) t ∈ [0 , ∞ )-martingale? Solution: (1) Since X t is bounded from above by tM t (and from below by tS t , where S t = inf s ≤ t B s ), we have | X t | ≤ t ( M t- S t ) . Since both M t and S t have the same distribution as | B t | (by symmetry), they are both in L 2 , and, therefore, so it X t ....
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MathReview8 - Exam Midterm Page 1of 3 Date University of...

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