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Aufsatz / Paper in SCI-Expanded-Zeitschrift

A comparative linear mean-square stability analysis of Maruyama- and Milstein-type methods

Buckwar E., Sickenberger T.: A comparative linear mean-square stability analysis of Maruyama- and Milstein-type methods, in: Mathematics and Computers in Simulation, Volume 81, Number 6, Page(s) 1110-1127, 2011.

BibTeX

@ARTICLE{
title = {A comparative linear mean-square stability analysis of Maruyama- and Milstein-type methods},
type = {Aufsatz / Paper in SCI-Expanded-Zeitschrift},
author = {Buckwar, Evelyn and Sickenberger, Thorsten},
language = {EN},
abstract = {In this article we compare the mean-square stability properties of the θ-Maruyama and θ-Milstein method that are used to solve stochastic differential equations. For the linear stability analysis, we propose an extension of the standard geometric Brownian motion as a test equation and consider a scalar linear test equation with several multiplicative noise terms. This test equation allows to begin investigating the influence of multi-dimensional noise on the stability behaviour of the methods while the analysis is still tractable. Our findings include: (i) the stability condition for the θ-Milstein method and thus, for some choices of θ, the conditions on the step-size, are much more restrictive than those for the θ-Maruyama method; (ii) the precise stability region of the θ-Milstein method explicitly depends on the noise terms. Further, we investigate the effect of introducing partial implicitness in the diffusion approximation terms of Milstein-type methods, thus obtaining the possibility to control the stability properties of these methods with a further method parameter σ. Numerical examples illustrate the results and provide a comparison of the stability behaviour of the different methods.},
pages = {1110-1127},
publisher = {Elsevier Science B.V. (North-Holland), Amsterdam},
journal = {Mathematics and Computers in Simulation},
volume = {81},
number = {6},
issn = {0378-4754},
month = {2},
year = {2011},
}

Details

Zusammenfassung: In this article we compare the mean-square stability properties of the θ-Maruyama and θ-Milstein method that are used to solve stochastic differential equations. For the linear stability analysis, we propose an extension of the standard geometric Brownian motion as a test equation and consider a scalar linear test equation with several multiplicative noise terms. This test equation allows to begin investigating the influence of multi-dimensional noise on the stability behaviour of the methods while the analysis is still tractable. Our findings include: (i) the stability condition for the θ-Milstein method and thus, for some choices of θ, the conditions on the step-size, are much more restrictive than those for the θ-Maruyama method; (ii) the precise stability region of the θ-Milstein method explicitly depends on the noise terms. Further, we investigate the effect of introducing partial implicitness in the diffusion approximation terms of Milstein-type methods, thus obtaining the possibility to control the stability properties of these methods with a further method parameter σ. Numerical examples illustrate the results and provide a comparison of the stability behaviour of the different methods.

Journal: Mathematics and Computers in Simulation
Volume: 81
Nummer: 6
Erscheinungsjahr: 2011
Seitenreferenz: 1110-1127
Anzahl Seiten: 18
Verlag: Elsevier Science B.V. (North-Holland), Amsterdam
ISSN: 0378-4754
Reichweite: International

Beteiligte

AutorInnen / HerausgeberInnen: Univ.-Prof. Dr. Evelyn Buckwar, Thorsten Sickenberger

Forschungseinheiten der JKU:

Wissenschaftszweige: 1103 Analysis | 1113 Mathematische Statistik | 1114 Numerische Mathematik | 1118 Wahrscheinlichkeitstheorie | 1121 Operations Research | 1145 Zeitreihenanalyse | 1165 Stochastik | 5943 Risikoforschung

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