Applied Probability

Applied Probability is an area which develops techniques for the use in stochastic modelling.

In this course we use concepts and tools from Markov processes, renewal theory, random walks and (optionally) themes like Markov additive processes and regeneration.

The class of phase-type distributions, defined in terms of absorption times in Markov processes, will play a major role througout the course. They constitute a class of distributions which may approximate any positive distribution arbitrarily close, and they provide for elegant solutions to complex problems by using probabilistic arguments often relying on sample path arguments and leading to explicit formulae expressed in terms of matrices. 

Their interplay with ladder height methods (in random walks), provide important applications e.g. ruin theory in  non-life insurance, where also Markov additive processes (Markov modulation) may be used, and in queueing theory regarding waiting time distributions. 

Though phase-type distributions are light-tailed, appropriate transformations give rise to dense classes of heavy-tailed distributions with e.g. Pareto, Weibull or Mittag-Leffler type of tails, in which basic distributional properties can be expressed in terms of functions of matrices again. These can be used in a similar way as phase-type distributions, and will be employed in the modelling of extremal events (e.g. insurance claims). 

The fitting of phase-type and/or their heavy-tailed counterpart to data will also be considered as this constitues an important part of their applicability.

The course is self-contained, providing all necessary background from both theory, applications and estimation. Students, who come around topics they have encountered previously, will benefit from reviewing the material in this alternative and highly probabilistic context, enabling a deeper understanding of the underlying subjects.