On continuous-time autoregressive fractionally integrated moving average processes

TL;DR

This paper introduces a continuous-time ARFIMA model driven by fractional Brownian motion, analyzing its stationarity, covariance, and spectral density to better understand long and short memory in time series.

Contribution

It extends ARFIMA models to continuous time, providing new theoretical insights into their stationarity and spectral properties.

Findings

Derived the covariance structure of CARFIMA

Established conditions for stationarity of the model

Computed the spectral density function for CARFIMA processes

Abstract

In this paper, we consider a continuous-time autoregressive fractionally integrated moving average (CARFIMA) model, which is defined as the stationary solution of a stochastic differential equation driven by a standard fractional Brownian motion. Like the discrete-time ARFIMA model, the CARFIMA model is useful for studying time series with short memory, long memory and antipersistence. We investigate the stationarity of the model and derive its covariance structure. In addition, we derive the spectral density function of a stationary CARFIMA process.