Unit name | Stochastic Processes |
---|---|
Unit code | MATHM6006 |
Credit points | 10 |
Level of study | M/7 |
Teaching block(s) |
Teaching Block 1A (weeks 1 - 6) |
Unit director | Dr. Yu |
Open unit status | Not open |
Pre-requisites |
None |
Co-requisites |
None |
School/department | School of Mathematics |
Faculty | Faculty of Science |
This course will begin with an introduction to Brownian motion. Starting from scratch, we will define Brownian motion and study its relation to the scaling of random walks, as well as several of its basic properties. We will then study several applications and extensions of Brownian motion. Among the topics we hope to include are: higher dimensional Brownian motions, the Brownian bridge and excursion, the fundamental relation to harmonic functions and differential equations, conformal invariance of Brownian motion, Ito's formula.
Aims
The aim of the unit is to introduce theory of Brownian motions, in particular, how to construct it from random walks, various properties, and finally stochastic integration leading to a brief survey of diffusion processes.
Syllabus
Existence and Explicit Construction of Brownian motion
Elementary properties (Markov property, reflection principle, hitting times)
Sample path properties (zero set, nowhere differentiability)
Stochastic integral and Ito's formula
Stochastic differential equations and Brownian bridge
Relation to Other Units
This unit is a first course in continuous time stochastic processes.
At the end of the unit students should:
Transferable Skills:
Understanding the behaviour of diffusion processes so as to be able to use them (e.g. perform calculations and write simulations) in problems arising in physics, engineering or statistics.
Lectures supported by problem sheets and solution sheets.
The assessment mark for Stochastic Processes:
Geoffrey R. Grimmett and David R. Stirzaker, Probability and Random Processes, Oxford University Press, 3rd edition can be used as the main reference on Brownian motion.
B. Oksendal, Stochastic Differential Equations: An Introduction with Applications, Springer, 6th edition can be used as a reference for stochastic integration.