Numerical Analysis of Explicit One-Step Methods for Stochastic Delay Differential Equations

Baker, Christopher T. H.; Buckwar, Evelyn

doi:10.1112/s1461157000000322

Cited by 146 publications

(102 citation statements)

References 17 publications

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“…Finally we show the implications of this resonance on the fully nonlinear system. The numerical simulations have used both the Euler-Maruyama method [28] and the multi-step method for stochastic delay differential equations [29]. By using a higher order method, we verify that the oscillations observed in the simulation are not due to the numerical method.…”

Section: Numerical Simulations and Resonant Behaviormentioning

confidence: 98%

Noise-Sensitivity in Machine Tool Vibrations

Buckwar

Kuske

Lespérance

et al. 2006

Int. J. Bifurcation Chaos

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We consider the effect of random variation in the material parameters in a model for machine tool vibrations, specifically regenerative chatter. We show that fluctuations in these parameters appear as both multiplicative and additive noise in the model. We focus on the effect of additive noise in amplifying small vibrations which appear in subcritical regimes. Coherence resonance is demonstrated through computations, and is proposed as a route for transitions to larger vibrations. The dynamics also exhibit scaling laws observed in the analysis of general stochastic delay differential models.

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Section: Numerical Simulations and Resonant Behaviormentioning

confidence: 98%

Noise-Sensitivity in Machine Tool Vibrations

Buckwar

Kuske

Lespérance

et al. 2006

Int. J. Bifurcation Chaos

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“…See also [20], page 227, [15] and [4]. As in the case of SODEs, the Cauchy-Maruyama scheme for SFDEs has order of convergence 1 2 ( [20], page 227, [15,4,8,14]). In Sections 2-5, we establish the strong Milstein scheme for SDDEs with several delays.…”

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confidence: 99%

Discrete-time approximations of stochastic delay equations: The Milstein scheme

Hu¹,

Mohammed²,

Yan³

2004

Ann. Probab.

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In this paper, we develop a strong Milstein approximation scheme for solving stochastic delay differential equations (SDDEs). The scheme has convergence order 1. In order to establish the scheme, we prove an infinitedimensional Itô formula for "tame" functions acting on the segment process of the solution of an SDDE. It is interesting to note that the presence of the memory in the SDDE requires the use of the Malliavin calculus and the anticipating stochastic analysis of Nualart and Pardoux. Given the nonanticipating nature of the SDDE, the use of anticipating calculus methods in the context of strong approximation schemes appears to be novel. Introduction.Discrete-time strong approximation schemes for stochastic ordinary differential equations (SODEs) are well developed. For an extensive study of these numerical schemes, one may refer to [17], [18] and [19], Chapters 5 and 6. Some basic ideas of strong and weak orders of convergence are illustrated in [13].If the rate of change of a physical system depends only on its present state and some noisy input, then the system can often be described by a stochastic ordinary differential equation (SODE). However, in many physical situations the rate of change of the state depends not only on the present but also on the past states of the system. In such cases, stochastic delay differential equations (SDDEs) or stochastic functional differential equations (SFDEs) provide important tools to describe and analyze these systems. For various aspects of the qualitative theory of SFDEs the reader may refer to [20,21] and the references therein.SDDEs and SFDEs arising in many applications cannot be solved explicitly. Hence, one needs to develop effective numerical techniques for such systems. Depending on the particular physical model, it may be necessary to design strong L p (or almost sure) numerical schemes for pathwise solutions of the underlying SFDE. Strong approximation schemes for SFDEs may be used to

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“…For stochastic FDEs the situation is much less satisfactory. A theorem concerning mean-square convergence for explicit one-step schemes applied to SFDEs with discrete delays and global Lipschitz coefficient functions has been presented in [2] (see also the references in that article for previous work on the topic). However, the main method used and investigated is the EulerMaruyama method, i.e.…”

Section: A Brief Review Of Methods and Aimsmentioning

confidence: 99%

“…the stochastic version of the basic Euler scheme. The consistency analysis in [2] was performed for the Euler-Maruyama method. The latter has also been applied to SFDEs with variable delays and local Lipschitz conditions on the coefficient functions, using an interpolation at non-meshpoints by piecewise constants, in [15].…”

Section: A Brief Review Of Methods and Aimsmentioning

confidence: 99%