MathReview8 - Exam: Midterm Page: 1of 3 Date: Mar 12, 2009...

Info iconThis preview shows pages 1–2. Sign up to view the full content.

View Full Document Right Arrow Icon

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

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 = ( 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 -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 ....
View Full Document

Page1 / 3

MathReview8 - Exam: Midterm Page: 1of 3 Date: Mar 12, 2009...

This preview shows document pages 1 - 2. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online