Faculty of Science

Applied Probability
Provider: Faculty of Science

Activity no.: 7007-23-07-31
	Enrollment deadline: 06/02/2023

Place

Department of Mathematical Sciences

Date and time

06.02.2023, at: 08:00 - 14.04.2023, at: 16:00

Regular seats

ECTS credits

7.50

Contact person

Nina Weisse E-mail address: weisse@math.ku.dk

Enrolment Handling/Course Organiser

Mogens Bladt E-mail address: bladt@math.ku.dk

Written language

English

Teaching language

English

Semester/Block

Block 3

Scheme group

Exam form

Oral examination, 30 minutes

Exam details

Oral examination with 30 min. preparation.

Grading scale

7 point grading scale. For PhD students: Passed / Not Passed

Course workload

Course workload category	Hours
Lectures	28.00
Preparation	163.00
Exercises	14.00
Exam	1.00

Sum	206.00

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.

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.

Literature

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

Teaching and learning methods

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

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