ILAS Seminar-E2

Numbering Code U-LAS70 10002 SE50 Year/Term 2021 ・ First semester
Number of Credits 2 Course Type seminar
Target Year Mainly 1st year students Target Student For all majors
Language English Day/Period Thu.4
Instructor name Croydon, David Alexander (Research Institute for Mathematical Sciences Associate Professor)
Outline and Purpose of the Course This seminar-style course will give students a chance to learn about some important models in applied probability. The focus will be on Markov chains, which are central to the understanding of random processes, and have applications in simulation, economics, optimal control, genetics, queues and many other areas. As well as introducing mathematical techniques, it will be a goal to show how these can be applied to understand certain random phenomena, such as the long-time behaviour of random walks, survival/extinction of branching processes, convergence of algorithms, and reinforcement.
Course Goals - To understand basic models of applied probability, particularly Markov chains
- To apply mathematical techniques to understand random phenomena in applications
- To gain experience in reading and presenting mathematics in English
Schedule and Contents In the first lecture, the lecturer will introduce the topic, and basic aims of the course. In subsequent weeks, students will be asked to prepare and present part of the material from one of the main textbooks, or their attempts to solve problems from these. The following indicates possible topics, though this may vary depending on the students’ proficiency level and background.

(1) Introduction to applied probability and Markov chains [1 week]
Review of basic probability, definition of a Markov chain, outline of course
(2) Discrete-time Markov chains [7 weeks]
Class structure, hitting times/probabilities, recurrence/transience, invariant distributions, convergence to equilibrium, time reversal, ergodic theorem
(3) Continuous-time Markov chains [3 weeks]
Generator, jump chain and holding times, Poisson processes
(4) Applications [3 weeks]
Random walks, branching processes, urn models, queuing models

Total: 14 classes and 1 week for feedback
Evaluation Methods and Policy Students will be expected to participate in class, both by presenting material prepared in advance, and by discussing problems. Their performance in these aspects will contribute 70% of the final mark. There will also be a written assignment, in which students will be asked to apply the techniques covered on the course, which will also contribute 30% of the final mark.
Course Requirements None
Study outside of Class (preparation and review) As noted in the course schedule, from the second week, students will be asked to prepare and present part of the material from one of the main textbooks, or their attempts to solve problems from these. (Their efforts on such assignments forms part of the assessment.) Details will depend on the number of students enrolled on the course, and will be discussed in the first class. Typically the lecturer would expect students to spend 1-2 hours per week on study outside the class.
Textbooks Textbooks/References Markov Chains, Norris, (University Press, 1997)
Probability and random processes, Grimmett and Stirzaker, (Oxford University Press, 2001)