HWK3 - IEOR E4701 Question 1 Assignment 3 Summer 2011...

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Unformatted text preview: IEOR E4701 Question 1 Assignment 3 Summer 2011 Consider the Markov chain whose transition probabilities are given by P = 0 1 2 0 1 2 1 0 2 1 2 1 2 0 3 3 3 1 0 4 4 a) b) c) Is the chain irreducible? why? Is the chain positive recurrent? why? Compute the equilibrium distribution in two dierent ways. 0 1/3 3/4 2/3 1/4 1/2 1/2 1 2 Question 2 In the context of credit risk models one often studies the distribution of default time using intensity based models. This question explores the construction of these models. Let (t) = 0 ^t (s) ds, t0 where (s) is a continuous non-negative function. Let N () be a Poisson process with rate 1. Consider M (t) = N ( (t)) t 0. M (t) can also be interpreted as an arrival process of customers. Let T1 = min {t 0 : M (t) = 1}, that is, T1 is the time of the rst arrival. Provide a formula for P (T1 > t). a) b) Suppose that X () is a continuous time Markov chain independent of N (). Let 1 IEOR E4701 Assignment 3 t 0 Summer 2011 (s) ds. Assume X (0) = i and (t) = r (X (t)) for some r > 0. Again, let (t) = provide an expression for P (T1 > t| X (0) = i). c) Derive an ODE that when solved, allows you to compute P (T1 < t| X (0) = i) . Question 3 Let {X (t) : t 0} be a continuous time Markov chain with the following transition rate matrix 0 A= 0 1 2 3 1 2 3 -4 1 2 1 2 -5 1 2 1 1 -3 1 1 1 0 -2 For a set D dene TD = min {t 0 : X (t) D}. Let B = {1} and C = {3}. Compute the following expectations: a) g (i) = Ei TB 0 f (X (s)) ds for i = 1 , where f (x) = sin x. T 0 b) h (i) = Ei exp r (X (s)) ds f (X (T )) where T = min {TB , TC }, f (x) = sin x and r (x) = x 100 . 2 ...
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HWK3 - IEOR E4701 Question 1 Assignment 3 Summer 2011...

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