*Course Syllabus for*
# Mathematical Foundations of Probability

Sannolikhetsteorins matematiska grunder

## MATM30F, 7.5 credits

**Valid from:** Autumn 2018

**Decided by:** Professor Thomas Johansson

**Date of establishment:** 2018-11-15

## General Information

**Division:** Mathematics

**Course type:** Course given jointly for second and third cycle

**The course is also given at second-cycle level with course code:** MATM30

**Teaching language:** English

## Aim

The aim of the course is to give mathematical knowledge that facilitates successful studies in probability, stochastic processes and stochastic differential equations. These subjects are becoming more and more important in many applications, such as meteorology, finance and social planning.

## Goals

*Knowledge and Understanding*

For a passing grade the doctoral student must

- be able to explain the measure theoretic approach to probabilities and random variables;
- be able to explain the construction of the Lebesgue-integral and the fundamental convergence theorem for this integral;
- be able to explain how the concepts conditional expectation and weak convergence can be formalized through measure theory.

*Competences and Skills*

For a passing grade the doctoral student must

- be able to use the fundamental theorems in integration theory to solve problems;
- be able to choose an appropriate solution strategy for a problem within the course's range, and thereafter work out a detailed solution.

## Course Contents

The course deepens and extends basic knowledge in probability theory. A central part of the course is existence and uniqueness theorems about measures defined on sigma-algebras, integration theory, conditional expectation and weak convergence in metric spaces.

## Course Literature

Probability. Springer Science & Business Media, 1996. ISBN 9780387945491.

**Types of instruction:** Lectures, exercises

**Examination format:** Oral exam

**Grading scale:** Failed, pass

**Examiner:**

## Admission Details

**Assumed prior knowledge:** Linear algebra, calculus in one and several variables, and basic courses in Fourier analysis and mathematical statistics.

## Course Occasion Information

**Course coordinators:**

**Web page:** http://www.ctr.maths.lu.se/course/MATM30/