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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.
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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 nonnegative 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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 Summer '10
 KarlSigma
 Markov process, Markov chain, Continuoustime Markov process, Continuous Time Markov, Markov models, time Markov chain

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