Stat219sample_final_f08_sol

Stat219sample_final_f08_sol - Math136/Stat219 Fall 2008...

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Math136/Stat219 Fall 2008 Sample Final Examination Write your name and sign the Honor code in the blue books provided. You have 3 hours to solve all 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. Important note: If you wish to use a result that is contained in an Exercise in the course notes that was not assigned for homework or proved in lecture, you must prove the result yourself (i.e. you cannot just site the Exercise number.) You may consult the following materials while taking the exam: 1. Stat219/Math136 Lecture notes, Fall 2008 version (the required text) 2. Kevin Ross’s Lecture slides posted in Coursework 3. Homework problems and solutions posted in Coursework 4. Your own graded homework papers 5. Your own notes taken during lecture Use of any other material is prohibited and constitutes a violation of the Honor Code. This includes, but is not limited to: other texts (including optional and recommended texts), photocopying of texts or notes, materials from previous sections of Stat219/Math136, the internet, programming formulas or other results in a calculator or computer, consultation with anyone during the exam (except for the Instructors or Teaching Assistants). 1. (5x5) Consider a simple, symmetric random walk. That is, let ξ 1 , ξ 2 , . . . be a sequence of independent random variables with P ( ξ k = 1) = P ( ξ k = 1) = 1 / 2 for all k . Let S 0 = 0 and S n = n k =1 ξ k for n = 1 , 2 , . . . . Let F n = σ ( ξ 1 , . . . , ξ n ). For fixed positive integers a and b let τ = min { n 0 : S n / ( a, b ) } . You may assume that τ < a.s. a) Show that τ is an {F n } -stopping time. 1
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ANS: { τ = n } = { S 1 ( a, b ) , . . . , S n - 1 ( a, b ) , S n / ( a, b ) } ∈ F n since F n = σ ( S 1 , . . . , S n ) (because S n is defined through an invertible transformation of ξ 1 , . . . , ξ n ; see Corollary 1.2.17). This is enough to show that τ is a discrete time {F n } -stopping time. b) Show that { S 2 n n } is an {F n } -martingale. ANS: Clearly, {F n } is a filtration to which { S 2 n n } is adapted. Checking integrability, since ξ k ∈ { 1 , 1 } , we have | S n | ≤ n and so IE | S 2 n n | ≤ n 2 + n < for any fixed n . Finally using the independence and distributions of ξ n : IE [ S 2 n +1 ( n + 1) |F n ] = IE [( S n + ξ n +1 ) 2 |F n ] ( n + 1) = S 2 n + IE ( ξ 2 n +1 ) 2 S n IE ( ξ n +1 ) ( n + 1) = S 2 n + 1 + 0 ( n + 1) = S 2 n n c) Show that E ( τ N ) = E ( S 2 τ N ) for any fixed positive integer N . ANS: For any fixed N , τ N is an {F n } -stopping time (see Exercise 4.3.3). Consider the stopped process { S 2 n ( τ N ) n ( τ N ) } . By same reasoning in part (b) we have that | S 2 n ( τ N ) n ( τ N ) | ≤ N 2 + N for all n ; thus the stopped process is uniformly integrable. By the optional stopping theorem (Theorem 4.3.8), IE [ S 2 τ N τ N ] = IE [ S 2 0 0] = 0 , which yields the result.
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