Introductory description
This module runs in Term 1 and is only available for students with their home department in Statistics. It is delivered by the Mathematics Department under the module code MA4F7.
Prerequisites: ST318 Probability Theory OR MA359 Measure Theory.
In 1827 the Botanist Robert Brown reported that pollen suspended in water exhibit random erratic movement. This ‘physical’ Brownian motion can be understood via the kinetic theory of heat as a result of collisions with molecules due to thermal motion. The phenomenon has later been related in Physics to the diffusion equation, which led Albert Einstein in 1905 to postulate certain properties for the motion of an idealized ‘Brownian particle’ with vanishing mass:
 the path t>B(t) of the particle should be continuous
 the displacements B(s+t)B(s) should be independent of the past motion, and have a Gaussian distribution with mean 0 and variance proportional to t
Module web page
Module aims
The module studies the construction and properties of Brownian motion, a fundamental tool for modelling processes which evolve randomly in time. Brownian motion is used widely in many areas of pure and applied mathematics and in the last few decades it has become essential to the study of financial maths as a model of stock prices.
Outline syllabus
This is an indicative module outline only to give an indication of the sort of topics that may be covered. Actual sessions held may differ.
Topics discussed in this module include:
 construction of Brownian motion/Wiener process
 fractal properties of the path, which is continuous but still a rough, nonsmooth function
 description as a Gaussian process, an important class of models in machine learning
 description as a Markov process in terms of generators and semigroups
 the martingale property of Brownian motion and some aspects of stochastic calculus
 scaling properties and connection to random walk
 connection to the Dirichlet problem, harmonic functions and PDEs
 some generalizations, including e.g. geometric Brownian motion and fractional Brownian motion
Learning outcomes
By the end of the module, students should be able to:
 Use the martingale property of BM to derive advanced properties such as Wald's lemmas;
 Describe its construction and explain simple properties of Brownian Motion (BM);
 Understand BM as a continuous time and continuous state Markov process;
 Understand the embedding of random walks in Brownian motion and use it to derive convergence results;
 Translate properties of onedimensional BM to higher dimensions.
Indicative reading list
Peter Mörters and Yuval Peres, Brownian Motion, Cambridge University Press, 2010
René L. Schilling and Lothar Partzsch, Brownian motion: an introduction to stochastic processes, De Gruyter, 2014
Thomas M. Liggett, Continuous Time Markov Processes  An Introduction, AMS Graduate studies in Mathematics 113, 2010
View reading list on Talis Aspire
Subject specific skills
At the end of the module students will be able to :
 describe its construction and explain simple properties of Brownian Motion (BM);
 understand BM as a continuous time and continuous state Markov process;
 use the martingale property of BM to derive advanced properties such as Wald’s lemmas;
 understand the embedding of random walks in Brownian motion and use it to derive convergence results;
 translate properties of onedimensional BM to higher dimensions.
Transferable skills
Students will acquire key reasoning and problem solving skills which will empower them to address new problems with confidence.
Study time
Type 
Required 
Optional 
Lectures 
30 sessions of 1 hour (20%)

2 sessions of 1 hour

Tutorials 
9 sessions of 1 hour (6%)


Private study 
111 hours (74%)


Total 
150 hours 

Private study description
Review lectured material and work on set exercises.
Costs
No further costs have been identified for this module.
You do not need to pass all assessment components to pass the module.
Students can register for this module without taking any assessment.
Assessment group D1

Weighting 
Study time 
Assignments worth 15%

15%


Coursework

Oncampus Examination

85%


The examination paper will contain five questions, of which the mark from the FIRST question and the best marks of THREE of the remaining four questions will be used to calculate your grade.
~Platforms  Moodle
 Answerbook Gold (24 page)

Assessment group R

Weighting 
Study time 
Online Examination

100%


The examination paper will contain five questions, of which the mark from the FIRST question and the best marks of THREE of the remaining four questions will be used to calculate your grade.
~Platforms  Moodle

Feedback on assessment
Marked coursework and exam feedback
Past exam papers for ST403
Courses
This module is Optional for:

Year 1 of
TMAAG1PE Master of Advanced Study in Mathematical Sciences

Year 1 of
TIBSN3G1 Postgraduate Taught Financial Mathematics

Year 1 of
TMAAG1P9 Postgraduate Taught Interdisciplinary Mathematics

Year 1 of
TMAAG1P0 Postgraduate Taught Mathematics

USTAG300 Undergraduate Master of Mathematics,Operational Research,Statistics and Economics

Year 3 of
G300 Mathematics, Operational Research, Statistics and Economics

Year 4 of
G300 Mathematics, Operational Research, Statistics and Economics
This module is Option list A for:

Year 4 of
USTAG300 Undergraduate Master of Mathematics,Operational Research,Statistics and Economics

Year 5 of
USTAG301 Undergraduate Master of Mathematics,Operational Research,Statistics and Economics (with Intercalated

Year 4 of
USTAG1G3 Undergraduate Mathematics and Statistics (BSc MMathStat)

Year 5 of
USTAG1G4 Undergraduate Mathematics and Statistics (BSc MMathStat) (with Intercalated Year)
This module is Option list B for:

Year 4 of
UCSAG4G3 Undergraduate Discrete Mathematics
This module is Option list D for:

Year 4 of
USTAG300 Undergraduate Master of Mathematics,Operational Research,Statistics and Economics

Year 5 of
USTAG301 Undergraduate Master of Mathematics,Operational Research,Statistics and Economics (with Intercalated
This module is Option list E for:

Year 4 of
USTAG300 Undergraduate Master of Mathematics,Operational Research,Statistics and Economics

Year 5 of
USTAG301 Undergraduate Master of Mathematics,Operational Research,Statistics and Economics (with Intercalated