Long Memory in a Linear Stochastic Volterra Differential Equation

TL;DR

This paper analyzes a linear stochastic Volterra equation with a stationary solution, demonstrating how the autocovariance function's decay rate relates to the kernel's properties, revealing long memory or subexponential decay.

Contribution

It establishes the precise asymptotic behavior of the autocovariance function based on the kernel's regular variation and log-convexity, extending understanding of long memory in such equations.

Findings

Autocovariance function is regularly varying at infinity.

Stationary solutions exhibit long memory or subexponential decay.

Decay rates can be arbitrarily slow under certain conditions.

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.