Syllabus:

# Stochastic Processes, 7.5 Credits

Swedish name: Stokastiska processer

This syllabus is valid: 2018-08-20 and until further notice

Course code: 5MS065

Credit points: 7.5

Education level: Second cycle

Main Field of Study and progress level: Mathematical Statistics: Second cycle, has only first-cycle course/s as entry requirements
Statistics: Second cycle, has only first-cycle course/s as entry requirements
Computational Science and Engineering: Second cycle, has only first-cycle course/s as entry requirements

Responsible department: Department of Mathematics and Mathematical Statistics

Established by: Faculty Board of Science and Technology, 2018-10-04

## Contents

The course gives an introduction to the theory of stochastic processes, especially Markov processes, and a basis for the use of stochastic processes as models in a great number of fields of application, such as queuing theory, Markov Chain Monte Carlo (MCMC), hidden Markov models (HMM) och financial mathematics. The course also covers simulation of stochastic processes and inference for the models. Furthermore, the course covers discrete Markov chains and Markov processes, the Markov property, Chapman-Kolmogorov's theorem and classification of Markov processes. The notions transition probability, transition intensity, forward- and backward equations, and stationary and asymptotic distributions are defined. The convergence of Markov chains, birth- and death processes, absorption probabilities, absorption times, renewal theory, martingales, and Brownian motion and diffusion are studied. Finally, an introduction to stochastic integration and stochastic differential equations is given.

## Expected learning outcomes

For a passing grade, the student must be able to

Knowledge and understanding
• thoroughly account for the theory of stochastic processes, especially Markov processes
• define Markov chains in discrete and continuous time, and classify a chain with respect to state, recurrence and transience, periodicity and irereducibility
• thoroughly account for Markov processes with continuous state space, especially Brownian motion and diffusion processes, and explain the connection between the theories of Markov processes and differential equations
• thoroughly describe the Markov Chain Monte Carlo (MCMC) method, and hidden Markov models (HMM)
Skills and abilities
• independently make calculations with transition probabilities and transition intensities
• decide the existence and uniqueness of stationary and asymptotic distributions of Markov chains, and if applicable derive such as solutions to equilibrity equations
• derive probabilities of absorption, and expected time until absorption for Markov chains
• set up appropriate Markov model, and conduct suitable calculations for different applications, especially when it comes to modelling with birth- and death processes
Judgement and approach
• value and compare different models from a scientific perspective

## Required Knowledge

The course requires a total of 90 ECTS including a course in Probability Theory on advanced level minimum 7,5 ECTS. Proficiency in English equivalent to Swedish upper secondary course English 5/A. Where the language of instruction is Swedish, applicants must prove proficiency in Swedish to the level required for basic eligibility for higher studies.

## Form of instruction

The teaching is mainly through lectures, problem solving sessions and supervision.

## Examination modes

The course is examined by a written exam. For the course, one of the following grades is assigned: Fail (U), Pass (G), Pass with distinction (VG). The grade is decided by the written exam.

A student who has received a passing grade on a test is not allowed to retake the test in order to receive a higher grade. A student who has not received a passing grade after participating in two tests has the right to be assigned another examiner, unless there are certain circumstances prohibiting this (see the Higher Education Ordinance, Chapter 6, 22§). A request to be assigned another examiner should be addressed to the head of department for the department of mathematics and mathematical statistics. The possibility of being examined based on the current version of the syllabus is guaranteed for at least two years following the student's first participation in the course.

Credit transfer
All students have the right to have their previous education or equivalent, and  their working life experience evaluated for possible consideration in the corresponding education at Umeå university. Application forms should be adressed to Student services/Degree evaluation office. More information regarding credit transfer can be found on the student web pages of Umeå university, http://www.student.umu.se, and in the Higher Education Ordinance (chapter 6). If denied, the application can be appealed (as per the Higher Education Ordinance, chapter 12) to Överklagandenämnden för högskolan. This includes partially denied applications

## Other regulations

In a degree, this course may not be included together with another course with a similar content. If unsure, students should ask the Director of Studies in Mathematics and Mathematical Statistics. The course can also be included in the subject area of computational science and engineering.