# NMAK11011U Advanced Probability Theory 2 (VidSand2)

Volume 2020/2021
Education

MSc Programme in Actuarial Mathematics
MSc Programme in Mathematics
MSc Programme in Mathematics-Economics
MSc Programme in Statistics

Content
• Signed measures, absolute continuity and singularity of measures, the Radon-Nikodym Theorem.
• Conditional expectations given a sigma-algebra.
• Martingales and submartingales in discrete time, the martingale convergence theorem, stopping times and optional sampling.
• Central Limit Theorem for martingales.
• Brownian motion: definition, continuity, variation and quadratic variation, non-differentiability, the law of the iterated logarithm.
Learning Outcome

Knowledge:

Basic knowledge of the topics covered by the course:  Decompositions of signed measures, conditional expectations, martingale theory, CLT for martingales, and definition, existence and path properties of the Brownian motion.

Skill:

• describe and prove the results on decomposition of signed measures.
• use the calculation rules for conditional expectations.
• show whether a sequence of random variables is a martingale or a submartingale.
• derive and describe the main results on martingales.
• apply the results on martingales to concrete examples.
• describe the foundation for the construction of stochastic processes in continuous time.
• describe the basic properties of the sample paths for Brownian motion.

Competence:

• discuss the relation between decomposition of measures and conditional expectations.
• relate and compare the results on martingales.
• use martingale CLT and understand the result compared to the classical CLT.
• discuss the concept of sample paths with a view to constructing continuous stochastic processes.
• Give an oral presentation of a specific topic within the theory covered by the course.
Advances probability theory 1(VidSand1) or equivalent

Academic qualifications equivalent to a BSc degree is recommended.
5 hours of lectures and 4 hours of exercises per week for 7 weeks.
The course is equivalent to Stochastic Processes 3 (NMAB15026U)
• Category
• Hours
• Lectures
• 35
• Preparation
• 132
• Theory exercises
• 28
• Project work
• 10
• Exam
• 1
• Total
• 206
Written
Oral
Continuous feedback during the course of the semester
Feedback by final exam (In addition to the grade)

Written feedback in the form of comments to the compulsory
assignements.

Oral feedback during exercise classes, as a response to the
contribution of the students to the solution process of the
exercises.

Detailed oral feedback after the oral exam.

Credit
7,5 ECTS
Type of assessment
Oral examination, 30 minutes
30 min preparation. All aids allowed during preparation.
Exam registration requirements

To participate in the exam the compulsory assignment must be approved and valid.

Aid
All aids allowed
Marking scale
7-point grading scale
Censorship form
External censorship
Re-exam

As the ordinary exam. If the compulsory assignment was not approved before the ordinary exam it must be (re)submitted. It must be approved three weeks before the beginning of the re-exam week.

##### Criteria for exam assesment

The student must in a satisfactory way demonstrate that he/she has mastered the learning outcome.