Spectral Solution Method for Distributed Delay Stochastic Differential Equations

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Title: Spectral Solution Method for Distributed Delay Stochastic Differential Equations
Authors: René, Alexandre
Date: 2016
Abstract: Stochastic delay differential equations naturally arise in models of complex natural phenomena, yet continue to resist efforts to find analytical solutions to them: general solutions are limited to linear systems with additive noise and a single delayed term. In this work we solve the case of distributed delays in linear systems with additive noise. Key to our solution is the development of a consistent interpretation for integrals over stochastic variables, obtained by means of a virtual discretization procedure. This procedure makes no assumption on the form of noise, and would likely be useful for a wider variety of cases than those we have considered. We show how it can be used to map the distributed delay equation to a known multivariate system, and obtain expressions for the system's time-dependent mean and autocovariance. These are in the form of series over the system's natural modes and completely define the solution. — An interpretation of the system as an amplitude process is explored. We show that for a wide range of realistic parameters, dynamics are dominated by only a few modes, implying that most of the observed behaviour of stochastic delayed equations is constrained to a low-dimensional subspace. — The expression for the autocovariance is given particular attention. A recurring problem for stochastic delay equations is the description of their temporal structure. We show that the series expression for the autocovariance does converge over a meaningful range of time lags, and therefore provides a means of describing this temporal structure.
URL: http://hdl.handle.net/10393/34327
http://dx.doi.org/10.20381/ruor-5172
CollectionThèses, 2011 - // Theses, 2011 -
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