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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 σalgebra 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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 Fall '11
 Gilbert
 Math, Probability, Continuous function, Dominated convergence theorem

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