L Evy-driven Continuous-time Arma Processes

Gaussian ARMA processes with continuous time parameter, otherwise known as stationary continuous-time Gaussian processes with rational spectral density , have been of interest for many years. 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 diierential equation models to problems in nance, exempliied by the derivation of the Black-Scholes option-pricing formula and its generalizations (Hull and White (1987)). Numerous examples of econometric applications 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 Acker-son (1990)). Like their discrete-time counterparts, continuous-time ARMA processes constitute a very convenient parametric family of stationary processes exhibiting a wide range of autocorrelation functions which can be used to model the empirical autocorrelations observed in nancial time series analysis. In nancial applications it has been observed that jumps play an important role in the realistic modelling of asset prices and derived series such a s v olatility. This has led to an upsurge of interest in L evy processes and their applications to nancial modelling. In this article we discuss second-order L evy-driven continuous-time ARMA models, their properties and some of their nancial applications, in particular to the modelling of stochas-tic volatility in the class of models introduced by Barndorr-Nielsen and Shephard (2001) and to the construction of a class of continuous-time GARCH models which generalize the COGARCH(1,1) process of Kl uppelberg, Lindner and Maller (2004) and which exhibit properties analogous to those of the discrete-time GARCH(p:q) process.