ST403 Brownian Motion
ST40315 Brownian Motion
Introductory description
This module runs in Term 2 and is only available for students with their home department in Statistics.
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 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.
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 
Seminars  (0%)  
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 D2
Weighting  Study time  

Assignments worth 15%  15%  
Coursework 

Inperson 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.

Assessment group R1
Weighting  Study time  

Inperson Examination  Resit  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.

Feedback on assessment
Marked coursework and exam feedback
Antirequisite modules
If you take this module, you cannot also take:
 MA4F715 Brownian Motion
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
Catalogue 
Resources 
Feedback and Evaluation 
Grade Distribution 
Timetable 
Assessments dates for Statistics modules, including coursework and examinations, can be found in the Statistics Assessment Handbook.