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Stochastic Processes - MAT00018M

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  • Department: Mathematics
  • Module co-ordinator: Prof. Zdzislaw Brzezniak
  • Credit value: 10 credits
  • Credit level: M
  • Academic year of delivery: 2021-22
    • See module specification for other years: 2022-23

Related modules

Pre-requisite modules

Co-requisite modules

  • None

Prohibited combinations

Module will run

Occurrence Teaching period
A Autumn Term 2021-22
B Spring Term 2021-22

Module aims

  • To introduce students to a range of mathematical models that take account of the stochastic (random) fluctuations that are always present in the real world;
  • To demonstrate the circumstances in which continuous-time stochastic models give results that are different to those from deterministic models that neglect the random effects;
  • To provide a range of mathematical techniques and approximations that can be used to make analytic predictions from stochastic models;


Module learning outcomes

At the end of the module you should be able to...

  • appreciate the uses for stochastic models, their characteristics and limitations;
  • explain the concept of continuous-time stochastic processes and the Markov property;
  • give examples of applications of stochastic processes;
  • formulate and analyse Markov models in continuous time;
  • calculate (conditional) probabilities of events and expectations of variables described by simple Markov processes like the Poisson process or the Wiener process
  • determine transition rates and stationary distributions of birth-death processes
  • understand  the basic properties of the Wiener process;
  • define the Ito stochastic integral and give its important properties.
  • Apply the  Ito Lemma to  verify if certain processes are martingales and to  find solutions of certain stochastic differential equations;


Module content

  • Principles of stochastic modelling
    • The need for models
    • Stochastic vs. deterministic models.
    • Continuous-time stochastic processes
    • The relevance of the Markov property
  • Discrete state-space: Markov jump processes
    • Poisson process
    • Birth-death processes
    • Kolmogorov equations (master equations)
    • Stationary distributions
  • Continuous state-space, stochastic calculus
    • Wiener process
    • Stochastic differential equations and Ito’s calculus. Examples include: geometric Brownian motion;
    • Stochastic integration

Coursework Assessment: Students will be required to complete directed private study with assessed coursework. Students will complete:

  • A reading study on discrete-time Markov chains, with a detailed assignment for a particular application and a written report on a topical system.


Task Length % of module mark
N/A 25
Online Exam -less than 24hrs (Centrally scheduled)
Stochastic Processes
2 hours 75

Special assessment rules



Task Length % of module mark
Online Exam -less than 24hrs (Centrally scheduled)
Reassessment: Examination - 2hrs -Stochastic Processes
2 hours 100

Module feedback

  • Online feedback on coursework
  • Discussion of coursework in seminars following due date
  • Typeset model solutions to exercise problems
  • Examination result delivered in Week 10 of Spring Term, with model solutions and examiner’s comments available earlier.

Indicative reading

  • G R Grimmett & D R Stirzaker, Probability and random processes, OUP.
  • C W Gardiner, Handbook of stochastic methods, Springer.
  • T Mikosch, Elementary stochastic calculus with finance in view, World Scientific.
  • Z Brzezniak & T Zastawniak, Basic Stochastic Processes, Springer.


The information on this page is indicative of the module that is currently on offer. The University is constantly exploring ways to enhance and improve its degree programmes and therefore reserves the right to make variations to the content and method of delivery of modules, and to discontinue modules, if such action is reasonably considered to be necessary by the University. Where appropriate, the University will notify and consult with affected students in advance about any changes that are required in line with the University's policy on the Approval of Modifications to Existing Taught Programmes of Study.