Spectral Solution Method for Distributed Delay Stochastic Differential Equations

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.