ST202 Stochastic Processes

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Lecturer(s): Dr Nick Tawn

Prerequisite(s):

• Statistics students: ST115 Introduction to Probability and MA137 Mathematical Analysis
• Non-Statistics students: ST111/112 Probability A & B and either MA131 Analysis I or MA137 Mathematical Analysis.

Leads to: ST333 Applied Stochastic Processes.

Commitment: 3 lectures/week, 1 tutorial each in weeks 3, 5, 7 and 9. This module runs in Term 2.

Content: Loosely speaking, a stochastic or random process is something which develops randomly in time. Only the simplest models will be considered in this course, namely those where the process moves by a sequence of jumps in discrete time steps. We will discuss: Markov chains, which use the idea of conditional probability to provide a flexible and widely applicable family of random processes; random walks, which serve as fundamental building blocks for constructing other processes as well as being important in their own right; and renewal theory, which studies processes which occasionally “begin all over again.” Such processes are common tools in economics, biology, psychology and operations research, so they are very useful as well as attractive and interesting theories.

Aims: To introduce the idea of a stochastic process, and to show how simple probability and matrix theory can be used to build this notion into a beautiful and useful piece of applied mathematics.

Objectives: At the end of the course students will:

• understand the notion of a Markov chain, and how simple ideas of conditional probability and matrices can be used to give a thorough and effective account of discrete-time Markov chains;
• understand notions of long-time behaviour including transience, recurrence, and equilibrium;
• be able to apply these ideas to answer basic questions in several applied situations including genetics, branching processes and random walks.

Assessment: 90% by 2 hour examination, 10% by coursework.

Deadlines: Assignment 1: week 2, Assignment 2: week 4, Assignment 3: week 6 and Assignment 4: week 8.

Examination period: Summer

Feedback: You will hand in answers to selected questions on the fortnightly exercise sheets. Your work will be marked and returned to you in the tutorial taking place the following week when you will have the opportunity to discuss it