# Mathematics

## Random processes

Module code: 862G1
Level 7 (Masters)
15 credits in spring teaching
Teaching method: Lecture
Assessment modes: Unseen examination, Coursework

Topics covered include:

Rationalisation:
After the introduction of the Poisson process, birth and death processes as well as epidemics models can be presented in full generality as applications of the pooled Poisson process. At the same time, the students will be introduced to the Kolmogorov equations and to the techniques for solving them. Renewal theory is needed to better understand queues, and, for this reason, it is discussed before queues.
Modernisation:
A modern introductory course on stochastic processes must include at least a section on compound renewal processes (with a focus on the compound Poisson process) as well as a chapter on the Wiener process and on Ito stochastic calculus. This is necessary given the importance this process has in several applications from finance to physics. Modernisation is achieved by including a new introductory chapter divided into three parts.
1. Poisson processes:
1. Density and distribution of into-event time.
2. Pooled Poisson process.
3. Breaking down a Poisson process.
4. Applications of the Poisson process, eg birth-and-death processes, the Kolmogorov equations.
1. Renewal processes
1. The ordinary renewal process.
2. The equilibrium renewal process.
3. The compound renewal process.
4. Applications of renewal processes, queues.
1. Wiener process
1. Definition and properties
2. Introduction to stochastic integrals
3. Introduction to stochastic differential equations.

### Pre-requisite

Pre-requisite: Level 6: (G1100) Probability Models [T1]

### Module learning outcomes

• Systematic understanding of the assumptions underlying continuous time models and how the models are derived.
• Develop advanced skills to 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.
• Comprehensive understanding of the Markov property and developing the ability to identify when it applies, and be able to analyse the models and apply them to different examples.