final-sol - Mathematics 136 Final Winter 2008 Write your...

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Mathematics 136, Final Winter 2008 Write your name and sign the Honor code in the blue books provided. This is a closed material exam, but you may use six pages (2 sides each) of notes. You have 180 minutes to solve all four questions, each worth points as marked (maximum of 100). Complete reasoning is required for full credit. You may cite lecture notes and homework sets, as needed, stating precisely the result you use, why and how it applies. 1. (5x5) Provide the complete, accurate and rigorous definitions for the following five concepts. (a) Pointwise convergence, almost sure convergence, and convergence in probability of a sequence of random variables { X n } on a probability space (Ω , F , P ) to another R.V. X on this space. ANS: Pointwise convergence happens when X n ( ω ) X ( ω )for each ω Ω. For a.s. con- vergence and convergence in probability detail Definitions 1.3.1 and 1.3.5. (b) A version of a stochastic process X t , a modification of a stochastic process X t and the canon- ical filtration of X t . ANS: Definitions 3.1.8, 3.1.9 and 4.1.3. (c) A right-continuous filtration F t , a square-integrable martingale ( M t , F t ) with continuous sam- ple path, and the increasing process A t associated with it. ANS: Definitions 4.2.1 for a martingale ( M t , F t ) and 4.2.10 for a right-continuous filtration. The S.P. { M t } is square-integrable if E M 2 t is finite for all t and has continuous sample path if t 7→ M t ( ω ) is a continuous function for all ω Ω. For the increasing process A t associated with it detail Theorem 4.4.7 and Definition 4.4.8. (d) The Brownian motion W t and the Geometric Brownian motion Y t . ANS: The Geometric Brownian motion is Y t = e W t and for the Brownian motion detail Definition 5.1.1. (e) A Markov chain { X n } , a homogeneous Markov chain and its strong Markov property. ANS: Detail Definition 6.1.1, Definition 6.1.2 and (6.1.1). 2. (4+4+4+9+4+4) Parts a)–f) of this problem are independent and can be solved independently of each other! Consider independent Brownian motions X t and Y t , the filtration F t = σ ( X s , Y s , s t ) and the process R t = X 2 t + Y 2 t . (a) Which, if any, of the stochastic processes X t and R t is a Gaussian process and which if any is a stationary process? ANS: From Definition 5.1.1 we know that X t is a Gaussian process. Note that R 1 0 and Var( R 1 ) > 0. Such R.V. can not be Gaussian (check Proposition 3.2.10), so the process { R t } is not Gaussian. Further, E R t = 2 E X 2 t = 2 t is non-constant, so both processes are non-stationary. 1
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(b) Compute the probability that X 2 > X 1 . ANS: This is the probability that the increment X 2 - X 1 is positive. Since the latter is Gaussian of zero mean and positive variance, the probability in question is 1 / 2.
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