# ST403 Brownian Motion

Please note that all lectures for Statistics modules taught in the 2022-23 academic year will be delivered on campus, and that the information below relates only to the hybrid teaching methods utilised in 2021-22 as a response to Coronavirus. We will update the Additional Information (linked on the right side of this page) prior to the start of the 2022/23 academic year.

Throughout the 2021-22 academic year, we will be adapting the way we teach and assess your modules in line with government guidance on social distancing and other protective measures in response to Coronavirus. Teaching will vary between online and on-campus delivery through the year, and you should read the information made available for MA4F7 about how this module will be delivered in 20/21. The contact hours shown in the module information below are superseded by the additional information.You can find out more about the University’s overall response to Coronavirus at: https://warwick.ac.uk/coronavirus.

All dates for assessments for Statistics modules, including coursework and examinations, can be found in the Statistics Assessment Handbook at http://go.warwick.ac.uk/STassessmenthandbook

# ST403-15 Brownian Motion

21/22
Department
Statistics
Level
Oleg Zaboronski
Credit value
15
Module duration
10 weeks
Assessment
Multiple
Study location
University of Warwick main campus, Coventry

##### 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 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, non-smooth 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 one-dimensional BM to higher dimensions.

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

##### 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 one-dimensional 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

On-campus 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

##### Assessment group R
Weighting Study time
In-person 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.

~Platforms - Moodle

##### Feedback on assessment

Marked coursework and exam feedback

## Courses

This module is Optional for:

• Year 1 of TMAA-G1PE Master of Advanced Study in Mathematical Sciences
• Year 1 of TIBS-N3G1 Postgraduate Taught Financial Mathematics
• Year 1 of TMAA-G1P9 Postgraduate Taught Interdisciplinary Mathematics
• Year 1 of TMAA-G1P0 Postgraduate Taught Mathematics
• USTA-G300 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 USTA-G300 Undergraduate Master of Mathematics,Operational Research,Statistics and Economics
• Year 5 of USTA-G301 Undergraduate Master of Mathematics,Operational Research,Statistics and Economics (with Intercalated
• Year 4 of USTA-G1G3 Undergraduate Mathematics and Statistics (BSc MMathStat)
• Year 5 of USTA-G1G4 Undergraduate Mathematics and Statistics (BSc MMathStat) (with Intercalated Year)

This module is Option list B for:

• Year 4 of UCSA-G4G3 Undergraduate Discrete Mathematics

This module is Option list D for:

• Year 4 of USTA-G300 Undergraduate Master of Mathematics,Operational Research,Statistics and Economics
• Year 5 of USTA-G301 Undergraduate Master of Mathematics,Operational Research,Statistics and Economics (with Intercalated

This module is Option list E for:

• Year 4 of USTA-G300 Undergraduate Master of Mathematics,Operational Research,Statistics and Economics
• Year 5 of USTA-G301 Undergraduate Master of Mathematics,Operational Research,Statistics and Economics (with Intercalated