Introductory description

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

Prerequisites:

• UG Students: ST318 Probability Theory OR ST350 Measure Theory for Probability OR MA359 Measure Theory.

• MSc in Statistics Students: ST964 Introduction to Advanced Probability

• MSc in Mathematical Finance Students: ST908 Stochastic Calculus for Finance.

Module web page

Module aims

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:

• Apply the martingale property of Brownian Motion (BM) to derive advanced properties such as Wald's lemmas.
• Construct and explain properties of BM.
• Apply BM as a continuous time and continuous state Markov process.
• Apply the embedding of random walks in Brownian motion and use it to derive convergence results.
• Translate properties of one-dimensional BM to higher dimensions.

Indicative reading list

Reading lists can be found in Talis

Specific reading list for the module

Subject specific skills

• 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

• Problem solving: Use rational and logical reasoning to deduce appropriate and well-reasoned conclusions. Retain an open mind, optimistic of finding solutions, thinking laterally and creatively to look beyond the obvious. Know how to learn from failure.

• Self awareness: Reflect on learning, seeking feedback on and evaluating personal practices, strengths and opportunities for personal growth.

• Communication: Present arguments, knowledge and ideas, in a range of formats.

• Professionalism: Prepared to operate autonomously. Aware of how to be efficient and resilient. Manage priorities and time. Self-motivated, setting and achieving goals, prioritising tasks.