Long memory in a linear stochastic Volterra differential equation |
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Authors: | John AD Appleby |
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Institution: | a School of Mathematical Sciences, Dublin City University, Dublin 9, Ireland b Institut für Mathematik, Humboldt-Universität zu Berlin, Unter den Linden 6, 10099 Berlin, Germany |
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Abstract: | In this paper we consider a linear stochastic Volterra equation which has a stationary solution. We show that when the kernel of the fundamental solution is regularly varying at infinity with a log-convex tail integral, then the autocovariance function of the stationary solution is also regularly varying at infinity and its exact pointwise rate of decay can be determined. Moreover, it can be shown that this stationary process has either long memory in the sense that the autocovariance function is not integrable over the reals or is subexponential. Under certain conditions upon the kernel, even arbitrarily slow decay rates of the autocovariance function can be achieved. Analogous results are obtained for the corresponding discrete equation. |
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Keywords: | Volterra integro-differential equations Volterra difference equations Itô -Volterra integro-differential equations Differential resolvent Asymptotic stability Stationary solutions Long memory Long-range dependence Regular variation Subexponential |
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