Upon successful completion of the course, the student will be able to
Explain and apply the Lebesgue integral for positive measures, included product measures.
Explain the convergence theorems (the monotone case, Fatou’s lemma and the dominated case) as well as the Tonelli-Fubini theorem.
Apply measure- and integration theory at fundamental parts of probability theory.
Know the measure theoretic definitions of probability, independence, random variable, stochastic process and conditional expectation, as well as various ways of describing the probability distribution of random variables.
apply methods in order to treat and describe limits of sequences of random variables, in particular the law of great numbers, the central limit theorem and convergence of martingales.
Have knowledge of how filtrations and conditional expectations are used to represent information.
Treat martingales in discrete time.
Contents
Lebesgue integral when measure and sigma-algebra are given, convergence theorems, product measure and integration on product spaces. Probability described by measure theoretic concepts, with themes like distribution and density, filtrations, conditional expectations, convergence of seuences of random variables, laws of great numbers, the central limit theorem, martingales in discrete time and filtrations.
Teaching and learning methods
Lectures, group work, project with applications to industrial mathematics and compulsory submissions. Estimated workload of the course is 267 hours.
Examination requirements
Mandatory hand-ins must be approved. See Canvas for details.
Examinations
Individual, oral exam
Student evaluation
The person responsible for the course decides, in cooperation with student representative, the form of student evaluation and whether the course is to have a midway or end of course evaluation in accordance with the quality system for education, chapter 4.1.