Stochastic differential equations

Research output: Chapter in Book/Report/Conference proceedingChapter

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.

LanguageEnglish
Title of host publicationMathematical Tools for Physicists
EditorsMichael Grinfeld
Pages73-108
Edition2
Publication statusPublished - 2014

Fingerprint

Numerical Approximation
Stochastic Equations
Differential equation
Stochastic Integration
Weak Approximation
Itô's Formula
Colored Noise
Fokker-Planck Equation
White noise
Convert
Differentiable
Brownian motion
Exact Solution
Path
Formulation

Cite this

Lord, G. J. (2014). Stochastic differential equations. In M. Grinfeld (Ed.), Mathematical Tools for Physicists (2 ed., pp. 73-108)
Lord, Gabriel James. / Stochastic differential equations. Mathematical Tools for Physicists. editor / Michael Grinfeld. 2. ed. 2014. pp. 73-108
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author = "Lord, {Gabriel James}",
year = "2014",
language = "English",
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pages = "73--108",
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}

Lord, GJ 2014, Stochastic differential equations. in M Grinfeld (ed.), Mathematical Tools for Physicists. 2 edn, pp. 73-108.

Stochastic differential equations. / Lord, Gabriel James.

Mathematical Tools for Physicists. ed. / Michael Grinfeld. 2. ed. 2014. p. 73-108.

Research output: Chapter in Book/Report/Conference proceedingChapter

TY - CHAP

T1 - Stochastic differential equations

AU - Lord, Gabriel James

PY - 2014

Y1 - 2014

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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.

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BT - Mathematical Tools for Physicists

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Lord GJ. Stochastic differential equations. In Grinfeld M, editor, Mathematical Tools for Physicists. 2 ed. 2014. p. 73-108