Module MATH4091: STOCHASTIC PROCESSES IV

Department: MATHEMATICAL SCIENCES

MATH4091: STOCHASTIC PROCESSES IV

Prerequisites

• Linear Algebra II (MATH2021), Analysis in Many Variables II (MATH2031).

Corequisites

• None.

Excluded Combination of Modules

• Stochastic Processes III (MATH3251).

Aims

• This module continues on from the treatment of probability in MATH1012 (Core Mathematics A).
• It is designed to introduce mathematics students to the wide variety of models of systems in which sequences of events are governed by probabilistic laws.
• Students completing this course should be equipped to read for themselves much of the vast literature on applications to problems in physics, engineering, chemistry, biology, medicine, psychology and many other fields.

Content

• Infinite sample spaces, random variables, expectation, joint distributions, conditional probability and expectation, partition theorems.
• Discrete time, discrete state Markov chains.
• Poisson processes.
• Stationary Gaussian processes.
• Continuous time Markov chains.
• Topics chosen from: Stochastic dynamic programming, information theory, entropy and relative entropy, percolation theory and the contact process, Brownian motion.
• Reading material on a topic related to: Markov chains, random walks, applications for queuing.

Learning Outcomes

• By the end of the module students will: be able to solve complex, unpredictable and specialised problems in Stochastic Processes.
• have an understanding of specialised and complex theoretical mathematics in the field of Stochastic Processes.
• have mastered a coherent body of knowledge of these subjects demonstrated through one or more of the following topic areas: Probability.
• have an advanced understanding in one of the following areas: Markov chains, random walks.
• Renewal Theory.
• Discrete time Markov processes.
• Poisson processes.
• Stationary processes.
• Continuous time Markov processes.
• Branching processes.

• In addition students will have highly specialised and advanced mathematical skills in the following areas: Modelling, Computation.

• Students will be able to study independently to further their knowledge of an advanced topic.

Modes of Teaching, Learning and Assessment and how these contribute to the learning outcomes of the module

• Lectures demonstrate what is required to be learned and the application of the theory to practical examples.
• Subject material assigned for independent study develops the ability to acquire knowledge and understanding without dependence on lectures.
• Assignments for self-study develop problem-solving skills and enable students to test and develop their knowledge and understanding.
• Formatively assessed assignments provide practice in the application of logic and high level of rigour as well as feedback for the students and the lecturer on students' progress.
• The end-of-year examination assesses the knowledge acquired and the ability to solve complex and specialised problems. The Subject material assigned for independent study will form part of the examined material.

Teaching Methods and Learning Hours

Summative Assessment

Formative Assessment:

Four written assignments to be assessed and returned. Other assignments are set for self-study and complete solutions are made available to students.

■ Attendance at all activities marked with this symbol will be monitored. Students who fail to attend these activities, or to complete the summative or formative assessment specified above, will be subject to the procedures defined in the University's General Regulation V, and may be required to leave the University