Module Content

Introduction to stochastic processes: Basic terminology, Examples of stochastic processes, Definition of stochastic processes. 

Markov Chains: Definitions, Examples of Markov Chains, Classification of States of a Markov Chain, Recurrence, Discrete Renewal Equation, Absorption Probabilities, Criteria of Recurrence, Queuing Example, Random Walk. 

Continuous Time Markov Chains: Birth Processes and Poisson Processes, Birth and Death Processes. 

Renewal Processes: Definition and Examples of Renewal Processes, Renewal Theorem and Applications. 

Martingales: Definitions and Examples, Optional Sampling Theorem, Convergence Theorems. 

Brownian Motion: Joint Probabilities for Brownian Motion, Continuity of paths and the Maximum Variables, Variations and Extensions. 

Stationary Processes: Definitions and Examples, Mean Square Distance, Mean square error prediction, Prediction of Covariance Stationary Processes, Ergodic Theory and Applications, Gaussian Systems. 

Teaching Methods

The module will be delivered on campus, with weekly lecture and tutorial sessions. 

Printed notes will be given ahead of time for each section of the course, to support and enhance students’ preparation and engagement during class sessions. Lectures will follow the notes, with discussions of the main theoretical topics, and study of examples of the applications of the theory. There will be a strong emphasis on student involvement in discussions in lectures, to encourage a more active approach to learning the material, and to allow the delivery to be tailored to build on the students’ current understanding. 

Regular formative work in tutorial sessions will allow students to internalise the mathematical ideas and methods developed in the lectures, and lead to the development of problem-solving skills. This formative work will also feed back into the delivery of lectures and tutorials. 

Assessment Methods

The module is assessed through a Portfolio of exercises and an examination.