Lévy–Driven Continuous–Time ARMA Processes

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