MATH 452/STAT 552 Assignment 5 Due: December 2nd, 2020 at the beginning of class. Submission: Upload your solutions of Exercises 5.1–5.5 to Crowdmark as one PDF file. (Do not hand in any solution to the Supplementary Exercises.) Exercise 5.1. Given a PDF f on [0,∞), the renewal equation for the corresponding renewal function m(t) [1, (7.5) on p.435] states that m(t) = ∫ t 0 f(x)dx+ ∫ t 0 m(t− x)f(x)dx, t ≥ 0. From this identity, the following identity for the Laplace transform of m(t) was shown in class: ∫ ∞ 0 e−µtm(t)dt = µ−1 ∫∞ 0 e−µtf(t)dt 1− ∫∞ 0 e−µtf(t)dt , µ ∈ (0,∞). (5.1) Find the Laplace transform of the PDF of a hyperexponential distribution: f(x) =∑n j=1 Pjλje −λjx, x > 0, where (Pj) is a probability vector and λj > 0 [1, p.302]. Then use your result and (5.1) to get the explicit solution of the Laplace transform of the corresponding renewal function. Exercise 5.2. (1) Find the Laplace transform of the ruin probability R(x) defined on [1, p.489] by applying the derivation of (5.1) to [1, (7.53) on p.490]. Express your solution explicitly in R(0), λ, c, and the Laplace transform of F (x). (2) Apply your solution in (1) to the case where F is the CDF of an exponential random variable with mean λ. Exercise 5.3. Let B and B′ be two independent one-dimensional standard Brownian motions such that B0 = B′0 = 0. Define a stochastic process (Wt;−∞ < t < ∞) by setting Wt = Bt if t ≥ 0 and Wt = B′−t if t < 0. Show that W has independent increments: for all integers n ≥ 3 and −∞ < t1 < t2 < · · · < tn <∞, Wt2 −Wt1 , Wt3 −Wt2 , · · · ,Wtn −Wtn−1 are independent. In your verification, it is enough to include 0 as one of the time points tj. Moreover, show that Wt+h −Wt ∼ N (0, |h|), for all t, h ∈ R. For the next two exercises, recall that with respect to a one-dimensional standard Brownian motion B (starting from zero) and a ∈ R, we write Ta for the first hitting time of a. 1 Exercise 5.4. Fix a > 0. Show that by [1, (10.6) on p.643], the two random variables Ta and (a/B1)2 have the same CDF. Exercise 5.5. This exercise aims for an alternative proof of the identity in [1, p.644] by using martingales. Read [1, p.674] for the definition of martingales, Exercise 17 and the martingale stopping theorem. (1) Show that aP(Ta < Tb) + bP(Tb < Ta) = 0, ∀ a < 0 < b. (5.2) (2) Show that for a < 0 < b, ({Ta < Tb} ∪ {Tb < Ta}){ ⊂ {Tb = ∞}, and then use the explicit distribution of max0≤s≤tBs to show that P(Tb =∞) = 0. Therefore, P(Ta < Tb) + P(Tb < Ta) = 1. (5.3) (3) Conclude this exercise by using (5.2) and (5.3) to find P(Ta < Tb) and P(Tb < Ta). Supplementary Exercises: 2, 5 [1, Chapter 7]. 1, 3, 5, 10, 17, 18 [1, Chapter 10]. References [1] Ross, S. (2019). Introduction to Probability Models. 12th edition. Academic Press. 2
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