Unit information: Stochastic Processes in 2013/14

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

Description including Unit Aims

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.

Intended Learning Outcomes

At the end of the unit students should:

• be able to recall all definitions and main results,
• be able to understand on an intuitive level the reasoning behind proofs of major results,
• be able to apply the theory in standard situations,
• be able to use the ideas of the unit in unseen situations

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.

Teaching Information

Lectures supported by problem sheets and solution sheets.

Assessment Information

The assessment mark for Stochastic Processes:

• 100% by means of a standard closed book one and half hour examination in April consisting of THREE questions. A candidate's best TWO answers will be used for assessment. Calculators are NOT permitted in this examination.

Reading and References

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.