Stochastic differential equations with a fractionally filtered delay: a semimartingale model for long-range dependent processes

TL;DR

This paper introduces the stochastic fractional delay differential equation (SFDDE) model, which generates long-range dependent processes with semimartingale properties by applying fractional filtering to the drift, offering advantages over traditional noise-based methods.

Contribution

It presents the SFDDE model with existence, uniqueness, spectral analysis, and links to fractionally integrated CARMA processes, highlighting its novel approach to long-range dependence.

Findings

SFDDE solutions are semimartingales with hyperbolic autocovariance decay.

The model's local path behavior remains unaffected by long memory.

Connections to fractional CARMA processes are established.

Abstract

In this paper we introduce a model, the stochastic fractional delay differential equation (SFDDE), which is based on the linear stochastic delay differential equation and produces stationary processes with hyperbolically decaying autocovariance functions. The model departs from the usual way of incorporating this type of long-range dependence into a short-memory model as it is obtained by applying a fractional filter to the drift term rather than to the noise term. The advantages of this approach are that the corresponding long-range dependent solutions are semimartingales and the local behavior of the sample paths is unaffected by the degree of long memory. We prove existence and uniqueness of solutions to the SFDDEs and study their spectral densities and autocovariance functions. Moreover, we define a subclass of SFDDEs which we study in detail and relate to the well-known fractionally integrated CARMA processes. Finally, we consider the task of simulating from the defining SFDDEs.