Unit information: Introduction to Stochastic Analysis in 2018/19

Unit name	Introduction to Stochastic Analysis
Unit code	MATHM0032
Credit points	20
Level of study	M/7
Teaching block(s)	Teaching Block 1 (weeks 1 - 12)
Unit director	Dr. Yu
Open unit status	Not open
Pre-requisites	Either 1. Further Topics in Probability 3 (MATH30006), or 2. Probability 2 (MATH20008) and Measure Theory and Integration (MATH30007). From 2019/20 onwards, Applied Partial Differential Equations 2 (MATH20402) will also be a prerequisite.
Co-requisites	None.
School/department	School of Mathematics
Faculty	Faculty of Science

Description including Unit Aims

Unit aims

The aim of the unit is to introduce theory of Brownian motion, continuous martingales, stochastic integration, stochastic differential equations and diffusion processes. With particular emphasis on applications to physical sciences, financial mathematics and other branches of applied mathematics.

General Description of the Unit

The course is intended for Master's students with a sufficiently strong background in analysis. Construction and analytic properties of Brownian motion, stochastic integration, stochastic differential equations and their strong and weak solutions, various approaches to diffusion processes will be covered. These are all topics of central importance in the general advanced mathematical culture. Special emphasis will be put on various applications of the theory.

Intended Learning Outcomes

Learning Objectives

• To gain a good understanding of the basic notions and techniques of the theory of:
  • Brownian motion
  • Stochastic differential equations and their strong and weak solutions
  • Diffusion processes
  • Applications of these concepts.

• To prepare students for independent research in mathematics.

Teaching Information

Lectures supported by problem sheets and solution sheets.

Assessment Information

100% Examination.

Raw scores on the examinations will be determined according to the marking scheme written on the examination paper. The marking scheme, indicating the maximum score per question, is a guide to the relative weighting of the questions. Raw scores are moderated as described in the Undergraduate Handbook.

Reading and References

Essential

• Lecture notes and problem sheets

Further

• K.L. Chung, R. Williams: Introduction to stochastic integration. Second edition. Birkauser, 1989

• I. Karatzas, S. Shreve: Brownian Motion and Stochastic Calculus, Springer 1991

• F. Klebaner: Introduction to Stochastic Calculus With Applications, World Scientific, 2005

• J. Lamperti, Stochastic Processes: a Survey of the Mathematical Theory, Springer 1977

• B. Oksendal, Stochastic Differential Equations: An Introduction with Applications, 6th edition, Springer 2010