# STATS 325 : Stochastic Processes

## Science

### Course Prescription

Introduction to stochastic processes, including generating functions, branching processes, Markov chains, random walks.

### Course Overview

This course looks at the theory of stochastic processes, showing how complex systems can be built up from sequences of elementary random choices. In particular, it will present the theory and techniques of Markov chains which can be used as probability models in many diverse applications. The course may be useful for students with interests in Probability, Mathematics, Statistics, Operations Research, Finance, Engineering, Economics, and Theoretical Biology. Before taking this course students should have a good background in probability (at least Grade B in one of STATS125, STATS210, STATS225, or STATS320) as well as some mathematics (one of MATHS208, MATHS250, MATHS253, or equivalent). This course will also provide good preparation for more advanced courses in probability (such as STATS710, STATS720, or STATS723).

### Course Requirements

Prerequisite: 15 points from STATS 125, 210, 320, with at least a B pass, 15 points from MATHS 208, 250, 253 Restriction: STATS 721

### Capabilities Developed in this Course

 Capability 1: Disciplinary Knowledge and Practice Capability 2: Critical Thinking Capability 3: Solution Seeking Capability 4: Communication and Engagement Capability 5: Independence and Integrity

### Learning Outcomes

By the end of this course, students will be able to:
1. Apply the techniques and constructions of discrete and continuous time Markov chains to solve problems involving n-step transition probabilities, hitting probabilities, and stationary distributions. (Capability 1, 2 and 3)
2. Distinguish between transient and recurrent states in given finite and infinite Markov chains. (Capability 1 and 3)
3. Translate a concrete stochastic process into the corresponding Markov chain given by its transition probabilities or rates. (Capability 1, 2 and 3)
4. Apply generating functions to identify important features of Markov chains. (Capability 1 and 3)
5. Organise a complex calculation for coherence and readability. (Capability 1, 2, 3, 4 and 5)
6. Produce clearly reasoned proofs of general properties of Markov chains at an appropriate level of difficulty. (Capability 1, 2, 3, 4 and 5)

### Assessments

Assessment Type Percentage Classification
Coursework 30% Individual Coursework
Final Exam 60% Individual Examination
Test 10% Individual Test
1 2 3 4 5 6
Coursework
Final Exam
Test

A mark of at least 40% in the final exam and 50% overall is required to pass this course.

### Key Topics

Markov chains in discrete time: n-step transition probabilities, transition matrices. The Strong Markov Property. Recurrence and transience. Hitting probabilities. Stationary distributions and measures. Generating functions. Random walks. Branching processes. Markov chains in discrete space and continuous time: transition rates, infinitesimal generator, Q matrices, Kolmogorov equations. Resolvents and first hitting times. Equilibrium distributions. Time reversibility. Ergodic theorem.

### Learning Resources

"Markov Chains" by J.R.Norris (Cambridge University Press)

"Probability: an introduction" by  G.R.Grimmett and D. Welsh (Oxford University Press)

### Special Requirements

Attending tutorials to improve problem solving skills is highly recommended.

This course is a standard 15 point course and students are expected to spend 10 hours per week involved in each 15 point course that they are enrolled in.

For this course, each week you can typically expect 3 hours of lectures, a 1 hour tutorial, 2-3 hours of reading and thinking about the content, and 3-4 hours of work on assignments and/or test preparation.

### Digital Resources

Course materials are made available in a learning and collaboration tool called Canvas which also includes reading lists and lecture recordings (where available).

Please remember that the recording of any class on a personal device requires the permission of the instructor.

The content and delivery of content in this course are protected by copyright. Material belonging to others may have been used in this course and copied by and solely for the educational purposes of the University under license.

You may copy the course content for the purposes of private study or research, but you may not upload onto any third party site, make a further copy or sell, alter or further reproduce or distribute any part of the course content to another person.

The University of Auckland will not tolerate cheating, or assisting others to cheat, and views cheating in coursework as a serious academic offence. The work that a student submits for grading must be the student's own work, reflecting their learning. Where work from other sources is used, it must be properly acknowledged and referenced. This requirement also applies to sources on the internet. A student's assessed work may be reviewed against online source material using computerised detection mechanisms.

### Inclusive Learning

All students are asked to discuss any impairment related requirements privately, face to face and/or in written form with the course coordinator, lecturer or tutor.

Student Disability Services also provides support for students with a wide range of impairments, both visible and invisible, to succeed and excel at the University. For more information and contact details, please visit the Student Disability Services’ website at http://disability.auckland.ac.nz

### Special Circumstances

If your ability to complete assessed coursework is affected by illness or other personal circumstances outside of your control, contact a member of teaching staff as soon as possible before the assessment is due.

If your personal circumstances significantly affect your performance, or preparation, for an exam or eligible written test, refer to the University’s aegrotat or compassionate consideration page: https://www.auckland.ac.nz/en/students/academic-information/exams-and-final-results/during-exams/aegrotat-and-compassionate-consideration.html.

This should be done as soon as possible and no later than seven days after the affected test or exam date.

### Student Feedback

During the course Class Representatives in each class can take feedback to the staff responsible for the course and staff-student consultative committees.

At the end of the course students will be invited to give feedback on the course and teaching through a tool called SET or Qualtrics. The lecturers and course co-ordinators will consider all feedback.

Your feedback helps to improve the course and its delivery for all students.

### Student Charter and Responsibilities

The Student Charter assumes and acknowledges that students are active participants in the learning process and that they have responsibilities to the institution and the international community of scholars. The University expects that students will act at all times in a way that demonstrates respect for the rights of other students and staff so that the learning environment is both safe and productive. For further information visit Student Charter (https://www.auckland.ac.nz/en/students/forms-policies-and-guidelines/student-policies-and-guidelines/student-charter.html).

### Disclaimer

Elements of this outline may be subject to change. The latest information about the course will be available for enrolled students in Canvas.

In this course you may be asked to submit your coursework assessments digitally. The University reserves the right to conduct scheduled tests and examinations for this course online or through the use of computers or other electronic devices. Where tests or examinations are conducted online remote invigilation arrangements may be used. The final decision on the completion mode for a test or examination, and remote invigilation arrangements where applicable, will be advised to students at least 10 days prior to the scheduled date of the assessment, or in the case of an examination when the examination timetable is published.

Published on 03/07/2020 11:31 a.m.