*Course Syllabus for*
# Theory of Stochastic Processes

Teorin för stokastiska processer

## FMS015F, 7.5 credits

**Valid from:** Spring 2014

**Decided by:** FN1/Anders Gustafsson

**Date of establishment:** 2014-04-22

## General Information

**Division:** Mathematical Statistics

**Course type:** Third-cycle course

**Teaching language:** English

## Aim

The course is a natural continuation of the study of probability theory as a part of mathematical education in Lund University. The aim of the course is to provide introduction into the theory of
stochastic processes. The course gives necessary insight into the theory of stochastic processes
for PhD students in other areas that use stochastic processes in their research. The course is the basic part for further studies in the theory of stochastic processes and stochastic differential equation. With respect to the theory there is no overlap of this course with any other course given at the Mathematical center.

## Goals

*Knowledge and Understanding*

For a passing grade the doctoral student must

- have developed the ability for mathematical communication orally and in writing
- be familiar with basic concepts and methods in stochastic processes
- have acquired basic knowledge for further studies in mathematics and probability in particular.

## Course Contents

Stochastic integrals of deterministic functions. Shift operators Correlation functions.
Spectral representation. Infinite-dimensional distributions. Kolmogorov Theorem on ex-
tension. Markov moments, martingales. Markov processes, Markov properties and related
operators. Trajectories of Markov processes with continuous time. Infinitesimal operators.
Diffusion processes. Stochastic differential. Ito’s formula.

## Course Literature

Wentzell, A.D.: A Course in the Theory of Stochastic Processes.

Also Brownian Motion and Stochastic Calculus” av I. Karatzas and S. Shreve,
”Probability” av A.N. Shiryaev,
”Foundations of Modern Probability” av O. Kallenberg

**Types of instruction:** Lectures, seminars, exercises

**Examination format:** Oral exam.
Tatyana Turova

**Grading scale:** Failed, pass

**Examiner:**

## Admission Details

**Admission requirements:** 60 hp in Mathematics

**Assumed prior knowledge:** The students are expected to know at least some measure theory and probability theory

**Selection criteria:** The course is part of the main field of studies in Mathematical Statistics at the Faculty of Science. The course is optional at the Second cycle in a Masters degree in Mathematics or Statistics, or even for the doctoral studies. The course is also offered as a single subject course.

## Course Occasion Information

**Course coordinator:** Tatyana Turova `<tatyana.turova@matstat.lu.se>`