| Lecturer: | Dr Malwina Luczak | Professor Nick Bingham |
| Room: | B304 | B411 |
| Email: | M.J.Luczak@lse.ac.uk | Nick Bingham |
| Office hours: | Please see the office hours page | |
This is a half unit course, with lectures in the Lent Term. For timetabling information for this course, please see: http://www.lse.ac.uk/admin/timetables/confirmed/module_sessional/ma/30.htm
There are also revision lectures in the Summer Term.
Classes will start in Week 2 of the Lent Term and will continue into Week 1 of the Summer Term.
Reading
There are a number of books that you may find helpful:
Exercises will be set every week in the lectures. All handouts will appear on this web site at the appropriate time: if you miss one in a lecture, this is the first place to look. Homework should be handed in to the lecturers' homework box (on the ground floor of Columbia House). Please label your work with your name and course number. No late work will be accepted, but partial solutions or solutions to only some of the questions are welcome.
Marked work will be returned and discussed in classes.
The Calendar entry gives an overview of the course content.
Pre-requisites
Students should have attended either MA411 Probability and Measure, or ST409 Stochastic Processes, or equivalent. If you are intending to take the course, we encourage you to talk to us: we might be amenable to (reasonable!) suggestions about what material should be covered.
Learning outcomes
Having followed MA414, you should have acquired
Stochastic analysis is a branch of probability dealing with operations on stochastic processes. Its focus is to develop a consistent theory of integration for integrals of stochastic processes with respect to stochastic processes.
Stochastic analysis is used to model and analyse systems that behave randomly. Such systems may originate in the natural sciences: physics (e.g. Brownian motion, diffusion processes of particles subject to random forces), biochemistry (e.g. stochastic models of cellular chemical reaction networks), or ecology (e.g. models of competition between species). In the last 30 years or so, techniques from stochastic analysis have increasingly been used in financial mathematics to model the evolution in time of stock and bond prices.
The main focus of this course will be on Brownian motion. This is the most basic process, lying at the heart of stochastic analysis. It is sufficiently concrete and well-behaved that one can prove many things about it, as well as perform explicit calculations that are impossible for more complicated objects. At the same time, it can be used as a building block for large classes of more general stochastic processes, some of whose characteristics may then be analysed via Brownian motion. We will show existence and uniqueness for Brownian motion, and study its properties. A considerable amount of time will be devoted to rigorous study of the It
ô integral, an object central to stochastic calculus. Apart from the formal definition, we shall establish Itô's formula and martingale representation theorems, which are useful devices for explicit calculations of integrals. Another topic will be Girsanov's theorem, which says that altering slightly the drift of a wellbehaved Itô process will not affect the law of the process too dramatically. This theorem has many important applications in financial mathematics and economics.Finally, we are going to expand our scope to some more advanced stochastic processes and stochastic differential equations governing their evolution.
Course materials
Lecture notes and exercises will appear on the course materials page in due course. Comments, criticisms, suggestions for future development, and details of any errors and misprints should be sent to Dr Malwina Luczak.
Assessment is 100% based on a 2 hour formal examination in the Summer Term.