# ST403 Brownian Motion

**Being offered by Maths as MA4F7. Statistics students should register for ST403. **

** Prerequisite(s): **AT LEAST ONE OF: ST318 Probability Theory, MA359 Measure Theory.

**Commitment: **30 lectures and 9 support classes. This module runs in Term 1.

WHAT IS IT?

Brownian motion was originally the description given in physics for the random erratic movement of molecules. In 1905 Einstein made a detailed study in which he postulated certain properties should hold. In 1923 mathematical Brownian motion was born when a famous mathematician, Norbert Wiener, showed how to construct a random function *W*(*t*) giving the molecules “position” at time *t* which had Einstein’s properties.

WHY IS IT INTERESTING?

- It is a beautiful mathematical object worth studying both for its own sake and because of the deep links it has with other areas of mathematics, particularly in analysis.
- Brownian motion is a fundamental tool for modelling processes which evolve randomly in time. It is used widely in many areas of applied maths and in the last few decades it has become essential to the study of financial maths as a model of stock prices.

WHAT WILL WE LEARN?

**Construction.**According to Einstein

- the function *t* ↦ *W*(*t*) must be continuous – the molecule never jumps

- the displacement between times *s* and *t*, that is *W*(*t*) – *W*(*s*), should be independent of the past motion and its distribution should be Gaussian with mean zero and variance *t – s*.

We will investigate methods of constructing such random functions. It turns out the Gaussian distribution is essential – it is impossible to do with any other distribution.

**Properties of the paths.**The path*t*↦*W*(*t*) cannot be smooth. Look at http://users.stat.umn.edu/~geyer/Stoch/brown.html to see a simulation. The applet at this web site allows you to zoom in on a simulated path – notice it seems to look the same no matter how much it is magnified: Brownian motion is the ultimate fractal!**The stochastic calculus.**Ordinary calculus is a powerful method of doing calculations with smooth functions. As we have just seen Brownian paths are not smooth, but miraculously there is a “stochastic calculus” which was developed by a Japanese mathematician Ito in the 1940s and which allows us to do computations with Brownian motion.**Differential equations.**Differential equations are essential to modelling deterministic phenomena in applied maths and physics.

Somewhat surprisingly this can be solved probabilistically using Brownian motion – a fact that lies at the heart of the links between probability theory and analysis, and which is still today yielding new discoveries.

**Books: **

- Rene L. Schilling, Lothar Partzsch and Bjorn Bottcher, *Brownian Motion*, De Gruyter, 2014

- Peter Mörters and Yuval Peres, *Brownian Motion*, Cambridge University Press, 2010

Additional reading:

- Thomas M. Liggett, *Continuous Time Markov Processes - An Introduction*, AMS Graduate studies in Mathematics 113, 2010

- Rick Durrett, *Probability - Theory and Examples*, Cambridge University Press, 2019

** Assessment:** 15% by assignments, 85% by 3 hour examination in Summer

* Examination Period: *Summer