1. Markov chains and Markov jump processes.
Necessary background on Markov jump processes (Markov processes defined on a countable state-space) is provided, and will serve as the main basis for the rest of the course. In particular the concept of an intensity matrix (infinitesimal generator) shall play an important role.
2. Occupation times in Markov jump processes.
Using mainly transform methods, we shall obtain expressions for the joint distribution of the times spent in different states in a Markov jump process with finitely many states. Especially their expectations shall be of interest later in the course when dealing with the estimation of discretely observed Markov jump processes.
3. Phase-type distributions.
Phase-type distributions are defined in terms of Markov jump processes. They are the first exit times of a Markov jump process from a finite set of transient states. They provide a flexible and general class of distributions which enables the use of probabilistic arguments in complex stochastic models. They may approximate any positive distribution arbitrarily well (denseness) and many functionals of interest in risk theory and queueing theory may be calculated exactly by either obtaining an explicit formulae or through numerical procedures. We shall apply the phase-type distributions mainly to the area of risk theory and ruin probabilities.
4. Matrix-exponential distributions.
Phase-type distributions have a rational (i.e. a fraction between two polynomials) Laplace transform. The class of distributions with rational Laplace transforms is, however, larger than the class of phase-type distributions. They are often referred to as matrix-exponential distributions for reasons which will be apparent in the course. We analyze the class of matrix-exponential distributions and compare the methodology to that for phase-type distributions. We show that many results which are valid for phase-type distributions are also valid for matrix-exponential distributions.
5. Multivariate phase-type and matrix-exponential distributions.
We extend phase-type and matrix-exponential distributions to higher dimensions through varios method, and consider classes of multivariate distributions for dependent stochastic variables which have marginal distributions such as exponential, gamma or more general phase-type distributions.
6. Estimation of phase-type distributions.
In stochastic modeling, the data we may obtain for variables which are assumed to have a phase-type distribution are usually confined to being the absorption times. The paths of the Markov jump processes which generates the phase-type distributions are unobservable. From a statistical point of view we are hence dealing with incomplete data, and we shall provide two methods for dealing with such cases. The first method is the maximum likelihood approach which may use the EM algorithm for calculating the maximum likelihood estimates. The second method is via the Markov chain Monte Carlo methodology, which may also be seen as penalized likelihood. We also discuss implementation, use, advantages and drawbacks of either method.
7. Estimation of discretely observed Markov jump processes.
For Markov jump processes which runs in continuous time, we may in practice often confront the problem of only having knowledge abouts the state of the process at certain fixed time points. There may, hence, have been unobserved transitions between these fixed time points. Such a situation appears for example in the analysis of migration between credit ratings in the area of financial credit risk. Again we provide essentially two methods for the estimation of the transition rates through an EM algorithm and Markov chain Monte Carlo approach. We illustrate the method through a larger study of transition ratings based on real data. We establish methods for testing some of the basic assumption in the model.
