# Applied Probability

### Course content

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).

Education

MSc Programme in Actuarial Mathematics

Learning outcome

At the end of the course the student is expected to have:

The ability to employ the classical tools from Applied Probability for solving stochastic models by performing probabilistic (sample path) arguments.

Knowledge about renewal theory, random walks, Markov processes, phase-type distributions, ladder height distributions, ruin probabilities, severity of ruin, heavy-tailed modelling of extremal events.

Skills to formalize phase-type distributions and their transformed counterparts, discuss their theoretical background, and apply them in the modelling of risk and extremal events.

Competences to idenitify patterns of random phenomena and building adequate stochastic models which can be solved for by using Markov processes and related techniques.

7 weeks of lectures (2 x 2 hours per week) combined with theoretical and practical exercises (2 hours per week).

M. Bladt & B. F. Nielsen (2017) Matrix-exponential distributions in Applied Probability. Springer Verlag.

Probability theory at bachelors level, including measure theory. Some previous exposure to stochastic processes will be an advantage.

Academic qualifications equivalent to a BSc degree is recommended.

Continuous feedback during the course of the semester
ECTS
7,5 ECTS
Type of assessment
Oral examination, 30 min.
Oral examination with 30 min. preparation.
Aid
All aids allowed

During preparation only.

Marking scale
Censorship form
No external censorship
Several internal examiners.
##### Criteria for exam assessment

The student must in a satisfactory way demonstrate that he/she has mastered the learning outcome of the course.

Single subject courses (day)

• Category
• Hours
• Lectures
• 28
• Preparation
• 163
• Exercises
• 14
• Exam
• 1
• English
• 206