Titlebook: Matrix-Exponential Distributions in Applied Probability; Mogens Bladt,Bo Friis Nielsen Textbook 2017 Springer Science+Business Media LLC 2

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Matrix-Exponential Distributions in Applied Probability

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Preliminaries on Stochastic Processes,interest on their own, the aforementioned topics are essential for the construction of phase-type distributions, both discrete and continuous, as well as in applications, stochastic modeling, and statistical methods presented in the remainder of the book.

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Martingales and More General Markov Processes,s are, for example, seen in fair games in which both the gambler and the gambling house are expected to break even in the long run. Martingales, together with their extensions to sub- and supermartingales, appear all over in stochastic modeling, and they provide us with powerful tools and techniques

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Phase-Type Distributions,, for reasons that will become apparent in the next chapter, be referred to as matrix-exponential distributions. Within the class of matrix-exponential distributions there is the subclass of ., which are defined in terms of an underlying Markov jump process (or Markov chain). As opposed to a general

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Multivariate Distributions,te matrix-exponential distributions as the class containing all distributions with a joint transform that is a rational function. Before we analyze these multivariate matrix-exponential distributions in their full generality, we will discuss an important subclass based on linear rewards of the sojou

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Markov Additive Processes,both with or without Brownian components), Lévy processes, and Markov random walks. They can be used to define point processes such as the Markov modulated Poisson process, the Markovian arrival process (MAP), and Markov renewal processes. They seem particularly well suited in connection with matrix

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Some Applications to Risk Theory,r increments between claim arrivals. This model is of interest, not only because of the generality it offers for the specification of the premium function, but because it shows a general method for dealing with nonlinear processes that have phase-type (or matrix-exponential) jumps. The second theme