Mathematics

Random processes (L.6)

Module code: G1101
Level 6
15 credits in spring semester
Teaching method: Lecture
Assessment modes: Unseen examination, Coursework

The second module concerned with the notion of a random (stochastic) process, after the Probability Models module.

In this module you will expand the tools you learned in Probability Models to the study of such processes in continuous time (as opposed to discrete time).  The notion of continuous-time Markov chains in a discrete state space is central to the lecture and offers a wide range of applications. The first part of the module focuses on counting processes:  stochastic processes counting the number of occurrences of a type of events (number of buses arriving at a bus stop, number of births in a population).

In the last part of the module, stochastic processes valued in continuous state spaces are introduced, focusing on the family of Gaussian processes. The Brownian motion is a key example which is as important, by the range of its applicability, as the notion of normal variable.

The module will also help you develop further your modelling skills.

Questions may include:

• How can we model a certain problem using a continuous-time Markov chain or a Gaussian process?
• Can we understand how events are correlated at different times?
• Can the model be used to estimate probabilities, expected values, waiting times etc. If so, how?
• How can we understand what happens to the model over a very long interval of time?

Module learning outcomes

• Understand the assumptions underlying continuous time models and how the models are formed.
• Be able to analyse the models mathematically and to isolate the important factors.
• Know how to relate continuous time processes to discrete analogues and embedded processes.
• Understand the Markov property and be able to identify when it applies and be able to analyse the models and apply them to different examples.