NMAK19003U Applied Probability

Applied Probability is an area which develops techniques for the use in stochastic modelling. In this course we introduce some of the classical concepts and tools from Markov processes, renewal theory, random walks and (optionally) themes like Markov additive processes and regeneration. 

In particular, the class of phase-type distributions, defined in terms of absorption times in Markov processes, will play a predominant role, providing examples througout. In particular, their interplay with ladder height methods (in random walks), provides important applications towards ruin theory in  non-life insurance, where also Markov additive processes (Markov modulation) may be used. 

Phase-type distributions is a renowned class of distributions in Applied Probability, which allows for elegant solutions to complex problems through probabilistic arguments often relying sample path arguments and leading to explicit formulae expressed in terms of matrices.

Phase-type distributions, as well as their multivariate extension, give rise to dense classes of heavy-tailed distributions with e.g. Pareto, Weibull or Mittag-Leffler type of tails. They 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). 

Fitting of phase-type and/or their heavy-tailed counterpart may be considered.

The course is self-contained, providing all necessary background from both theory (stochastic processes) and applications (e.g. risk processes in insruance).