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Publication Type
Conference Proceedings
UWI Author(s)
Author, Analytic
Appleby, J.A.D; Kelly, C.
Author Role
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Author Affiliation
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Paper/Section Title
Almost Sure Asymptotic Behaviour of One- and Two-step Difference Equations with Random Coefficients on a Reducing Mesh
Medium Designator
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Editor/Compiler
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Proceedings Title
Proceedings of the IX International Chetayev Conference: Analytical Mechanics Stability, and Control of Motion
Date of Meeting
Refereed
Place of Meeting
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Place of Publication
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Publisher Name
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Date of Publication
2007
Date of Copyright
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Volume ID
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Series Editor
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Series Title
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Series Volume ID
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Location/URL
http:; www.dcu.ie/maths/research/preprints/ms0715.pdf
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Notes
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Abstract
We construct discrete models that mimic the asymptotic behaviour of a linear stochastic vanishing delay equation in circumstances where the solutions do not oscillate. It is known that solutions display oscillatory behaviour if the rate at which the delay vanishes is slow, and nonoscillatory behaviour if the rate is fast. It is possible to construct an auxiliary process, with identical oscillatory behaviour and differentiable sample paths, in order to develop a partial analysis of this behaviour with deterministic methods applied on a pathwise basis. In fact, in a previous paper we showed that a more complete description of the oscillatory behaviour could be gained using a two-step explicit Euler discretisation on a carefully chosen nonuniform mesh. In this paper we show that an equivalent explicit discretisation, and a closely related implicit discretisation, are consistent with the known asymptotic behaviour of the continuous process, in circumstances where the solutions are nonoscillatory. This behaviour can be stable or unstable in character. Therefore, we can use both first and second order difference equations with random coefficients to model the qualitative properties of a first order linear stochastic delay differential equation.....
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