Introductory description

We will introduce stochastic integration, and basic tools in stochastic analysis including Ito’s formula. We will also introduce lots of examples of stochastic differential equations.

Module web page

Module aims

Outline syllabus

This is an indicative module outline only to give an indication of the sort of topics that may be covered. Actual sessions held may differ.

• Construction of the Ito stochastic integral.

• Ito’s formula, stopping times, time changes.

• Existence and uniqueness for stochastic differential equations (SDEs).

• One dimensional examples: Brownian Bridges, Ornstein Uhlenbeck process,
  Geometric Brownian motion, Feller diffusion. Exit analysis for one dimensional diffusions.

• Links between SDEs and PDEs. Higher dimensional examples. Invariant measures.

• Topics from: conditioning diffusions; small noise asymptotics; ergodic theorems and
  central limit results for diffusions.

Learning outcomes

By the end of the module, students should be able to:

• Understanding of how to build a variety diffusions using Brownian motions, Ito’s stochastic integral and via stochastic differential equations.
• Ability to exploit Ito’s formula to extract statistical information about diffusions.
• Knowledge of a range of diffusions models that have been used in maths, physics and finance.

Subject specific skills

Stochastic calculus is now a standard tool in modelling randomness in both scientific modelling,
and more recently in financial modelling. This module has been taken by mathematicians,
statisticians, physicists and economists.

Transferable skills

Stochastic calculus is now a standard tool in modelling randomness in both scientific modelling,
and more recently in financial modelling. This module has been taken by mathematicians,
statisticians, physicists and economists.