# Courses in English at Department of Mathematics and Statistics on 2018-2019

**Bachelor level courses:**

**MATA271 Stochastic Models** (4 ECTS, Spring Term)

**Contents:** In this course we study mainly Markov chains. Besides investigating their properties, for example the behavoir as time goes to infinity, we consider many applications, among them:

- a simple weather forecast model,
- a discrete-time share price model,
- a model to describe the risk of cancer caused by radiation
- random walk as a special case of a Markov chain

Finally we discuss that Markov Chain Monte Carlo methods work because there is a generalized Law of Large Numbers behind.**Learning Outcomes**: After the course, the student:

- knows Markov chains and their properties
- has studied several models where Markov chains are used
- can decide whether a certain real world situation can be modelled by Markov chains
- can analyze Markov chain models and derive properties for the real world situation
- has learned about several Markov Chain Monte Carlo methods.

**Prerequisites: **Elementary probability on the level of MATA280 *Foundations of stochastics or *TILA12

**Advansed/ Master level studies:**

**MATS254 Stochastic process** (4 ECTS, Autumn Term)

**Contents:** The course gives an introduction into the theory of martingales and some applications. Martingales are one of

the most important classes of stochastic processes. They are widely used in stochastic modelling and in pure mathematics itself. The content of the course is:

* martingales

* Doob's optional stopping theorem

* Doob's martingale convergence theorem

* applications (Branching Processes and Kakutani's Dichotomy Theorem)**Learning Outcomes:** After completion of the course, the student

* can calculate conditional expectations

* can decide whether a stochastic process is a martingale

* knows the basic conditions under which a martingale converges

* can apply martingales in stochastic modelling**Prerequisites**: MATA280 Foundations of stochastics,

Recommended: Measure theoretic foundation of probability

(MATS260 Probability 1 or MATS112 Measure and Integration Theory 2)

**MATS262 Probability Theory 2** (5 ECTS, Autumn Term)

**Contents**: * Types of convergence of random variables and measures

* Sums of independent random variables

* Convolution of probability measures

* Law of large numbers

* Central limit theorem

Learning Outcomes: After the course

* the student knows the types of convergence of random variables and measures as well as their relations to each other,

* the student is familiar with the behaviour of sums of independent random variables, and knows the Law of large numbers and the Central limit theorem

* the student can identify (multidimensional) Gaussian distributions and describe their properties using characteristic functions

**MATS280 Risk Theory** (5 ECTS, Autumn term)

**Contents: Stochastic modeling of non-live insurance: Poisson process, risk process, ruin probabili**ties, Cramer-Lundberg bounds,heavy and light tails for claim size distributions.**Learning Autcomes: **After completing the course, the participant knows

* how to model the basic risk of an insurance company,

* how to compute and estimate ruin probabilities,

* the difference in between modelling small and frequent risks (light tails) and big and rare risks (heavy tails).**Prerequisites: **MATA280 Foundations of stochastics or TILA121 Probability or TILA1200 Probability 1.

**MATS352 Stochastic Analysis** (5 ECTS, Spring Term)

**Contents**: The course introduces basics of stochastic analysis. One cornerstone is the Brownian motion, probably one of the most important stochastic processes. The course will cover:

* definition of the Brownian motion, its construction, and basic properties

* Stochastic integrals as an extension of Riemann-integrals

* It's formula as an extension of the Taylor formula from calculus**Learning Outcomes:** The students understand basic properties of the Brownian motion and can verify some of them. They are familiar with the construction of stochastic integrals. The students are able to compute particular stochastic integrals and to apply Itô's formula in various situations.**Prerequisites:** Course MATS262 Probability 2 or similar. Recommended: MATS254 Stochastic processes or similar.

**MATS353 Stochastic Differential Equations** (4 ECTS, Spring Term)

**Contents:** Stochastic differential equations are a modern and important tool in stochastic modelling and have also applications within partial differential equations, harmonic analysis, and other areas of mathematics.

The course covers the following topics:

- existence and uniqueness of solutions to stochastic differential equations

- properties of solutions

- solving particular stochastic differential equations

- applications in Finance**Learning Outcomes: **

- the student understands the concept of a solution of a stochastic differential equation

- the student knows the theorems from the course about the existence and the behaviour of solutions

- the student can solve some stochastic differential equations**Prerequisites**: Probability theory on the level of MATS352 *Stochastic Analysis*.

**MATS260 Probability Theory 1** (5 ECTS, Spring term)

**Contents**: Basic concepts of probability:

* probability space

* independence of events

* random variables

* expectation and its basic properties

* independence of random variables

**Learning Outcomes:** The students are familiar with the concept of probability spaces, random variables, and independence.

They are able to describe simple stochastic phenomena within this framework and know important distributions. The notion of expected values along with the main theorems about integration is understood as extension of the Riemann integral.

The students are able to compute expected values based on discrete distributions and the Lebesgue measure on the real line.

**Prerequisites:** MATA280 Foundations of stochastics or TILA121 Probability or TILA1200 Probability

### TILS8000 Elements of statistical learning (5 ECTS)