Applications of Markov chains and convex sets in ridk management
Abstract
In this dissertation we examine the conditions under which a model with a large number of elements can be approximated by a new, simpler one, consisting of a smaller number of elements, without missing the qualitative features of the initial model. The certain conditions facilitate the analysis of large and complex systems that are widely used in studies. We apply our research in two important tools for credit risk management: credit ratings and credit scoringParticularly, we first introduce Markov Chains as a model of the stochastic evolution of credit ratings and then consider the conditions under which a Markov chain is lumpable. We review briefly the definition and characterization of an exactly lumpable Markov chain and through an example of a credit migration matrix we show that in general these conditions are not satisfied. We then introduce the concept of approximate lumpability, and we propose a procedure for finding a lumpable Markov chain that is the closest approximation of ...
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