Stochastic differential equations

We give an introduction to Brownian motion and illustrate that although
sample paths are continuous they are nowhere differentiable. This
leads to a discussion on white and coloured noise.
In \secref{sec:StochInt} we introduce stochastic integration and look
at the difference between \Ito and Stratonovich integrals. We examine
\Ito SDEs, in \secref{sec:itosde}, giving some motivating examples,
and use the \Ito formula to find some exact solutions. We then
consider Stratonovich SDEs and show how to convert between \Ito and
Stratonovich cases.
We discuss numerical approximation of both
\Ito and Stratonovich SDEs and discuss the multilevel Monte-Carlo
method for weak approximation in \secref{sec:sdenum}.
We conclude in \secref{sec:SDEPDE} by looking at the link between SDEs and partial
differential equations (PDEs) and introduce both the Fokker--Planck
and backwards Fokker--Planck equations.

Throughout the aim is to develop intuition by first examining simple,
one-dimensional examples before giving more general formulations and
results. Also included are some basic algorithms for numerical approximation.