## Abstract

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.

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.

Original language | English |
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Title of host publication | Mathematical Tools for Physicists |

Editors | Michael Grinfeld |

Pages | 73-108 |

Edition | 2 |

Publication status | Published - 2014 |