Stochastic Processes

The course can be part of the following programmes: 1) Mathematical Sciences, Master's Programme (N2MAT) and 2) Bachelor's Programme in Mathematics (N1MAT).

Admission to the course requires knowledge corresponding to the course MSG110 (Probability theory) as well as knowledge corresponding to at least one of the courses MVE140/MSA150 (Foundations of probability theory), TMS110/MSG300 (Markov theory), MSG800/MVE172 (Basic stochastic processes), or MVE135 (Random processes with applications).

Fundamentals of probability theory: sigma-algebras and probability measures. Modes of convergence for sequences of random variables. Conditional expectation. Martingales in discrete time, including convergence theorems. Stationary processes and ergodic theory. Random walk. Exchangeable processes.

On successful completion of the course the student will be able to:

• Define and explain fundamental concepts in measure theoretic probability theory.
• Define the most important modes of convergence for sequences of random variables, and explain how they relate to each other.
• Prove the existence of conditional expectation and compute it for concrete examples.
• Define, provide examples of, and solve problems about martingales in discrete time.
• Prove the most important convergence theorems for martingales in discrete time.
• Prove and explain the ergodic theorem for stationary processes.
• Prove and apply de Finetti's theorem.

Lectures

Written and/or oral examination.

If a student who has twice received a failing grade for the same examination component wishes to change examiner ahead of the next examination session, such a request should be made to the department in writing and should be approved by the department unless there are special reasons to the contrary (Chapter 6 Section 22 of the Higher Education Ordinance).

If a student has received a recommendation from the University of Gothenburg for study support for students with disabilities, the examiner may, where it is compatible with the learning outcomes of the course and provided that no unreasonable resources are required, decide to allow the student to sit an adjusted exam or alternative form of assessment.

In the event that a course has ceased or undergone major changes, students are to be guaranteed at least three examination sessions (including the ordinary examination session) over a period of at least one year, but no more than two years after the course has ceased/been changed. The same applies to internships and professional placements (VFU), although this is restricted to just one additional examination session.

The grading scale comprises Pass (G), Pass with distinction (VG) and Fail (U).

The course will be evaluated at the end of the course together with the students. This will be followed by an anonymous survey.

The results of and possible changes to the course will be shared with students who participated in the evaluation and students who are starting the course.