# Applied Probability

### Course content

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.

Education

MSc Programme in Actuarial Mathematics
MSc Programme in Statistics
MSc Programme in Mathematics-Economics

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.
Type of assessment details
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

### Kursusinformation

Language
English
Course number
NMAK19003U
ECTS
7,5 ECTS
Programme level
Full Degree Master
Duration

1 block

Placement
Block 3
Schedulegroup
C
Capacity
No limit
The number of seats may be reduced in the late registration period
Studyboard
Study Board of Mathematics and Computer Science
##### Contracting department
• Department of Mathematical Sciences
##### Contracting faculty
• Faculty of Science
##### Course Coordinator
• Mogens Bladt   (5-4872676a7a4673677a6e34717b346a71)