Numerical simulation of a linear stochastic oscillator with additive noise
type: Item
instance of: scholarly article
title: Numerical simulation of a linear stochastic oscillator with additive noise
zbMATH Open document ID: 1060.65007
author: Aslaug H. Strømmen Melbø
author: Desmond J. Higham
published in: Applied Numerical Mathematics
publication date: 2004-10-12T00:00:00Z
full work available at URL: https://strathprints.strath.ac.uk/164/
review text: Consider ordinary stochastic differential equations (linear oscillators) governed by \[ dX(t) = Y(t) dt \] \[ dY(t) = - X(t) dt + \sigma dW(t) \] perturbed by additive noise, where \(W(t), t \geq 0\) is a standard \(d\)-dimensional Wiener process (\(W(0)=0\), \(E [W(t)]=0\), \(E [W(t)]^2=t\)), and \(\sigma\) is a nonrandom real-valued constant as its noise intensity. Assume that \(X(0)=x_0=1\) and \(Y(0)=0\) are fixed throughout the paper. Suppose that one discretizes this type of equation with constants \(\sigma > 0\) by Euler-type methods along nonrandom equidistant partitions of time-intervals \([0,T]\), where \(\Delta\) is the nonrandom mesh size. The ability of such numerical methods to reproduce long-time features of the above stated linear stochastic oscillator is examined. It is shown that the standard forward Euler-Maruyama method fails to capture the correct second moment growth rate, whereas a customized extension of the partitioned Euler method behaves well in this respect. Such a qualitative failure of standard Euler-Maruyama methods applied to more general systems of linear equations with both additive and multiplicative noise has been demonstrated already in the book of \textit{H. Schurz} [Stability, stationarity, and boundedness of some implicit numerical methods for stochastic differential equations and applications, (Logos-Verlag, Berlin) (1997; Zbl 0905.60002)]; and the paper of \textit{H. Schurz} [ZAMM, Z. Angew. Math. Mech. 79, No. 6, 375--382 (1999; Zbl 0938.34072)] among other publications. Thus, the main new result of this paper can be seen that the authors show that the partitioned Euler method inherits an infinite oscillation property. A weaker oscillation result is also proved for a wide class of Itô-Taylor methods applied to this linear oscillator with \(x_0=1\) and \(y_0=0\).
reviewed by: Henri Schurz
Mathematics Subject Classification ID: 65C30
Mathematics Subject Classification ID: 34F05
Mathematics Subject Classification ID: 37H10
Mathematics Subject Classification ID: 60H10
Mathematics Subject Classification ID: 60H30
Mathematics Subject Classification ID: 65L20
Mathematics Subject Classification ID: 93E15
Mathematics Subject Classification ID: 60H35
zbMATH DE Number: 2107329
zbMATH Keywords: stochastic systems
zbMATH Keywords: stochastic oscillator
zbMATH Keywords: linear oscillator with additive noise
zbMATH Keywords: stochastic differential equations
zbMATH Keywords: discretization methods
zbMATH Keywords: partitioned Euler method
zbMATH Keywords: Euler-Maruyama method
zbMATH Keywords: Itô-Taylor methods
zbMATH Keywords: oscillation properties
zbMATH Keywords: second moment growth rate
zbMATH Keywords: Wiener process
zbMATH Keywords: stability
MaRDI profile type: Publication
OpenAlex ID: W1992881609
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DOI: 10.1016/J.APNUM.2004.02.003
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description: scientific article; zbMATH DE number 2107329