Stochastic Processes, 7.5 Credits

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 large number of application areas, such as queing theory, Markov chain Monte Carlo, hidden Markov models and financial mathematics. Simulation of stochastic processes and inference for the models, is also included.

Element 1 (6,5 ECTS): Theory and applications
Topics that are covered include discrete Markov chains and Markov processes, the Markov property, Chapman-Kolmogorov's theorem and the classification of Markov processes. The notions tranisition probability, transition intensity, forwaard and backward equations, and stationary and asymptotic distributions, are treated. Furthermore the convergence of Markov chains, birth-and-death processes, absorbtion probabilities, time to absorbtion, renewal theory, martingales, Brownian motion and diffusion processes are studied. Finally stochastic integration and stochastic differential equations are introduced.

Element 2 (1 ECTS): Simulation of stochastic processes with suitable statistical software
The part contains programming, data analysis and presentation of the results in written form.

Expected learning outcomes

For a passing grade, the student must be able to

Knowledge and understanding

• thoroughly describe the theory of stochastic processes, especially for Markov processes
• define Markov chains in discrete and continuous time
• define the existence and uniqueness of stationary and asymptotic distributions for Markov chains, and whenever applicable derive the distributions as solutions to balance equations
• thoroughly explain the meaning of Markov processes with continous state space, especially Brownian motion and diffusion processes
• critically describe the connection between the theory of Markov processes and differential equations

Skills

• classify Markov chains in dicrete and continuous time with respect to state diagrams, recurrence and trancience, states, periodicty and irreducibility
• conduct calculations with transition probabilities and transition intensities
• calculate absorbtion probabilities and the expected time to absorbtion for Markov chains
• choose a proper Markov model and conduct proper calculations for different applications, especially regarding the modeling of birth-and-death processes
• apply the Markov chain Monte Carlo method and hidden Markov models

Judgement and approach

• critically evaluate simulation results with respect to relevant measures

Required Knowledge

The course requires an advanced course in Probability Theory, minimum 7,5 ECTS or equivalent. 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 in element 1 takes the form of lectures and lessons. The teaching in element 2 takes the form of lab work.

Examination modes

Element 1 is assessed through a written examination. Element 2 is assessed through written lab reports. For element 1, one of the following grades is awarded: Fail (U), Pass (3), Pass with merit (4), Pass with distinction (5). For element 2, one of the following grades is awarded: Fail (U), or Pass (G). For the course as whole, one of the following grades is awarded: Fail (U), Pass (3), Pass with merit (4), Pass with distinction (5). The grade for the whole course is determined by the grade given for element 1. To pass the whole course, all elements must have been passed. The grade is only set once all compulsory elements have been assessed.

A student who has been awarded a passing grade for the course cannot be reassessed for a higher grade. Students who do not pass a test or examination on the original date are given another date to retake the examination. A student who has sat two examinations for a course or a part of a course, without passing either examination, has the right to have another examiner appointed, provided there are no specific reasons for not doing so (Chapter 6, Section 22, HEO). The request for a new examiner is made to the Head of the Department of Mathematics and Mathematical Statistics. Examinations based on this course syllabus are guaranteed to be offered for two years after the date of the student's first registration for 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

This course can not be included in a degree together with another course with similar contents. When in doubt, the student should consult the director of study at the department of mathematics and mathematical statistics.