Stochastic Runge-Kutta method for stochastic delay differential equations

Random e�ect and time delay are inherent properties of many real phenomena around us, hence it is required to model the system via stochastic delay di�erential equations (SDDEs). However, the complexity arises due to the presence of both randomness and time delay. The analytical solution of SDDEs is...

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Main Author:	Rosli, Norhayati
Format:	Thesis
Language:	English
Published:	2012
Subjects:	QA Mathematics
Online Access:	http://eprints.utm.my/id/eprint/31539/1/NorhayatiRosliPFS2012.pdf
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spelling	my-utm-ep.315392018-04-27T01:16:53Z Stochastic Runge-Kutta method for stochastic delay differential equations 2012-04 Rosli, Norhayati QA Mathematics Random e�ect and time delay are inherent properties of many real phenomena around us, hence it is required to model the system via stochastic delay di�erential equations (SDDEs). However, the complexity arises due to the presence of both randomness and time delay. The analytical solution of SDDEs is hard to be found. In such a case, a numerical method provides a way to solve the problem. Nevertheless, due to the lacking of numerical methods available for solving SDDEs, a wide range of researchers among the mathematicians and scientists have not incorporated the important features of the real phenomena, which include randomness and time delay in modeling the system. Hence, this research aims to generalize the convergence proof of numerical methods for SDDEs when the drift and di�usion functions are Taylor expansion and to develop a stochastic Runge{Kutta for solving SDDEs. Motivated by the relative paucity of numerical methods accessible in simulating the strong solution of SDDEs, the numerical schemes developed in this research is hoped to bridge the gap between the evolution of numerical methods in ordinary di�erential equations (ODEs), delay di�erential equations (DDEs), stochastic di�erential equations (SDEs) and SDDEs. The extension of numerical methods of SDDEs is far from complete. Rate of convergence of recent numerical methods available in approximating the solution of SDDEs only reached the order of 1.0. One of the important factors of the rapid progression of the development of numerical methods for ODEs, DDEs and SDEs is the convergence proof of the approximation methods when the drift and di�usion coe�cients are Taylor expansion that had been generalized. The convergence proof of numerical schemes for SDDEs has yet to be generalized. Hence, this research is carried out to solve this problem. Furthermore, the derivative-free method has not yet been established. Hence, development of a derivative{free method with 1.5 order of convergence, namely stochastic Runge{ Kutta, to approximate the solution of SDDEs with a constant time lag, r > 0, is also included in this thesis. 2012-04 Thesis http://eprints.utm.my/id/eprint/31539/ http://eprints.utm.my/id/eprint/31539/1/NorhayatiRosliPFS2012.pdf application/pdf en public phd doctoral Universiti Teknologi Malaysia, Faculty of Science Faculty of Science
institution	Universiti Teknologi Malaysia
collection	UTM Institutional Repository
language	English
topic	QA Mathematics
spellingShingle	QA Mathematics Rosli, Norhayati Stochastic Runge-Kutta method for stochastic delay differential equations
description	Random e�ect and time delay are inherent properties of many real phenomena around us, hence it is required to model the system via stochastic delay di�erential equations (SDDEs). However, the complexity arises due to the presence of both randomness and time delay. The analytical solution of SDDEs is hard to be found. In such a case, a numerical method provides a way to solve the problem. Nevertheless, due to the lacking of numerical methods available for solving SDDEs, a wide range of researchers among the mathematicians and scientists have not incorporated the important features of the real phenomena, which include randomness and time delay in modeling the system. Hence, this research aims to generalize the convergence proof of numerical methods for SDDEs when the drift and di�usion functions are Taylor expansion and to develop a stochastic Runge{Kutta for solving SDDEs. Motivated by the relative paucity of numerical methods accessible in simulating the strong solution of SDDEs, the numerical schemes developed in this research is hoped to bridge the gap between the evolution of numerical methods in ordinary di�erential equations (ODEs), delay di�erential equations (DDEs), stochastic di�erential equations (SDEs) and SDDEs. The extension of numerical methods of SDDEs is far from complete. Rate of convergence of recent numerical methods available in approximating the solution of SDDEs only reached the order of 1.0. One of the important factors of the rapid progression of the development of numerical methods for ODEs, DDEs and SDEs is the convergence proof of the approximation methods when the drift and di�usion coe�cients are Taylor expansion that had been generalized. The convergence proof of numerical schemes for SDDEs has yet to be generalized. Hence, this research is carried out to solve this problem. Furthermore, the derivative-free method has not yet been established. Hence, development of a derivative{free method with 1.5 order of convergence, namely stochastic Runge{ Kutta, to approximate the solution of SDDEs with a constant time lag, r > 0, is also included in this thesis.
format	Thesis
qualification_name	Doctor of Philosophy (PhD.)
qualification_level	Doctorate
author	Rosli, Norhayati
author_facet	Rosli, Norhayati
author_sort	Rosli, Norhayati
title	Stochastic Runge-Kutta method for stochastic delay differential equations
title_short	Stochastic Runge-Kutta method for stochastic delay differential equations
title_full	Stochastic Runge-Kutta method for stochastic delay differential equations
title_fullStr	Stochastic Runge-Kutta method for stochastic delay differential equations
title_full_unstemmed	Stochastic Runge-Kutta method for stochastic delay differential equations
title_sort	stochastic runge-kutta method for stochastic delay differential equations
granting_institution	Universiti Teknologi Malaysia, Faculty of Science
granting_department	Faculty of Science
publishDate	2012
url	http://eprints.utm.my/id/eprint/31539/1/NorhayatiRosliPFS2012.pdf
_version_	1747815828432093184

Stochastic Runge-Kutta method for stochastic delay differential equations

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