### Abstract

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.

Language | English |
---|---|

Title of host publication | Mathematical Tools for Physicists |

Editors | Michael Grinfeld |

Pages | 73-108 |

Edition | 2 |

Publication status | Published - 2014 |

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### Cite this

*Mathematical Tools for Physicists*(2 ed., pp. 73-108)

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*Mathematical Tools for Physicists.*2 edn, pp. 73-108.

**Stochastic differential equations.** / Lord, Gabriel James.

Research output: Chapter in Book/Report/Conference proceeding › Chapter

TY - CHAP

T1 - Stochastic differential equations

AU - Lord, Gabriel James

PY - 2014

Y1 - 2014

N2 - We give an introduction to Brownian motion and illustrate that althoughsample paths are continuous they are nowhere differentiable. Thisleads to a discussion on white and coloured noise. In \secref{sec:StochInt} we introduce stochastic integration and lookat 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 thenconsider Stratonovich SDEs and show how to convert between \Ito andStratonovich cases. We discuss numerical approximation of both\Ito and Stratonovich SDEs and discuss the multilevel Monte-Carlomethod for weak approximation in \secref{sec:sdenum}.We conclude in \secref{sec:SDEPDE} by looking at the link between SDEs and partialdifferential equations (PDEs) and introduce both the Fokker--Planckand backwards Fokker--Planck equations.Throughout the aim is to develop intuition by first examining simple,one-dimensional examples before giving more general formulations andresults. Also included are some basic algorithms for numerical approximation.

AB - We give an introduction to Brownian motion and illustrate that althoughsample paths are continuous they are nowhere differentiable. Thisleads to a discussion on white and coloured noise. In \secref{sec:StochInt} we introduce stochastic integration and lookat 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 thenconsider Stratonovich SDEs and show how to convert between \Ito andStratonovich cases. We discuss numerical approximation of both\Ito and Stratonovich SDEs and discuss the multilevel Monte-Carlomethod for weak approximation in \secref{sec:sdenum}.We conclude in \secref{sec:SDEPDE} by looking at the link between SDEs and partialdifferential equations (PDEs) and introduce both the Fokker--Planckand backwards Fokker--Planck equations.Throughout the aim is to develop intuition by first examining simple,one-dimensional examples before giving more general formulations andresults. Also included are some basic algorithms for numerical approximation.

M3 - Chapter

SN - 978-3-527-41188-7

SP - 73

EP - 108

BT - Mathematical Tools for Physicists

A2 - Grinfeld, Michael

ER -