Course category:
3rd year
Aims: When describing complex systems, the deterministic description in terms of differential equations and smooth functions is not always appropriate. Instead we need descriptions that can take account of the stochastic (random) fluctuations that are always present in the real world. The paradigm example is Brownian motion which leads to paths that are everywhere continuous but nowhere differentiable. This module provides a first introduction to the calculus required to deal with such phenomena.
Learning objectives: At the end of the module you should be able to:
- show that certain simple stochastic processes are martingales;
- give several equivalent definitions of Brownian motion;
- show that certain transformations of a Brownian motion are also Brownian motions;
- describe how Brownian motion arises as the limit of sums of random variables;
- simulate Brownian motion on a computer;
- outline the construction of the stochastic integral;
- use the martingale property of the stochastic integral;
- recognise a stochastic differential equation, and understand how different coefficients can lead to different properties of the solution;
- apply Ito's Lemma to find solutions of certain stochastic differential equations;
- use a computer to numerically solve stochastic differential equations.
Syllabus:
- Intuitive introduction to stochastic differential equations
- Numerical solution of stochastic differential equations
- Basic concepts from measure-theoretic probability theory
- Conditional expectation given a sigma field
- Martingales
- Properties of Brownian motion
- Stochastic integral and the Ito lemma
- Ito stochastic differential equations
- Some applications of stochastic differential equations
Recommended texts:
- * Z Brzezniak & T Zastawniak, Basic Stochastic Processes, Springer (S 9.3 BRZ).
- C W Gardiner, Handbook of Stochastic Methods, Springer.
- I Karatzas and S E Shreve, Brownian motion and Stochastic Calculus, Springer (U 1.36 KAR).
- R Durrett, Stochastic Calculus: A Practical Introduction (S 9.3 DUR)
- R Durrett, Probability: Theory and Examples (S 9 DUR)
- T Mikosch, Elementary Stochastic Calculus (with finance in view) (S 9.3 MIK)
Teaching:
- Lectures - 2 hours per week
- Problems Class / Computer Practical - 1 hour per week
Assessment: One and a half hour closed examination 90%
Coursework 10%. Note that coursework submitted after the advertised deadlines will be given a mark of zero.
Elective information:
This module is not available as an elective.
Prerequisites: