Lévy–Driven Continuous–Time ARMA Processes

Abstract

Gaussian ARMA processes with continuous time parameter, oth- erwise known as stationary continuous-time Gaussian processes with rational spectral density, have been of interest for many years. (See for example the papers of Doob (1944), Bartlett (1946), Phillips (1959), Durbin (1961), Dzha- pararidze (1970,1971), Pham-Din-Tuan (1977) and the monograph of Arató (1982).) In the last twenty years there has been a resurgence of interest in continuous-time processes, partly as a result of the very successful application of stochastic differential equation models to problems in finance, exemplified by the derivation of the Black-Scholes option-pricing formula and its gener- alizations (Hull and White (1987)). Numerous examples of econometric ap- plications of continuous-time models are contained in the book of Bergstrom (1990). Continuous-time models have also been utilized very successfully for the modelling of irregularly-spaced data (Jones (1981, 1985), Jones and Ack- erson (1990)). Like their discrete-time counterparts, continuous-time ARMA processes constitute a very convenient parametric family of stationary pro- cesses exhibiting a wide range of autocorrelation functions which can be used to model the empirical autocorrelations observed in financial time series anal- ysis. In financial applications it has been observed that jumps play an impor- tant role in the realistic modelling of asset prices and derived series such as volatility. This has led to an upsurge of interest in Lévy processes and their applications to financial modelling. In this article we discuss second-order Lévy-driven continuous-time ARMA models, their properties and some of their financial applications. Examples are the modelling of stochastic volatil- ity in the class of models introduced by Barndorff-Nielsen and Shephard (2001) and the construction of a class of continuous-time GARCH models which generalize the COGARCH(1,1) process of Klüppelberg, Lindner and Maller (2004) and which exhibit properties analogous to those of the discrete- time GARCH(p, q) process.