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Absorbing Markov Chains with Random Transition Matrices and Applications


In this thesis, we explore Markov chains with random transition matrices. Such chains are a development on classic Markov chains where the transition matrix is taken to be random. The intuition for this is that we may be interested in modeling phenomenas where the homogeneity assumption of classic Markov chains is invalid. We first use such chains to model credit risk. The randomness of the transition matrix is used to represent the randomness of the economy that underlies credit risk. With this, we model a portfolio of loans and the risk due to having a shared economy. We then proceed to explore theoretical properties of such chains with a focus on their asymptotic behavior. In the case of absorbing chains, we show that the the infinite product of independent and identically distributed random matrices must converge almost surely. We also introduce perturbed Markov chains as a special form of Markov chains with random transition matrices. Perturbed Markov chains are Markov chains with transition matrices of the form P+εQ where P is taken to be a constant matrix, 0<ε<1 is a constant and Q random with rows summing to 0. For perturbed Markov chains with P irreducible, we approximate the long-run fluctuation of such chains. As for perturbed Markov chains with P having absorbing states, we approximate the variance of the fundamental matrix and mean time to absorption. We also explore some applications. In the Moran process application, we study the impact of temporal randomness on the Moran process and derive an analytic result to calculate the probability of fixation.

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