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- PETER BROCKWELL, ERDENEBAATAR CHADRAA, ALEXANDER LINDNER
- 2006

A family of continuous-time generalized autoregressive conditionally heteroscedastic processes, generalizing the COGARCH(1, 1) process of Klüppelberg, Lindner and Maller [J. Appl. Probab. 41 (2004) 601–622], is introduced and studied. The resulting COGARCH(p, q) processes, q ≥ p ≥ 1, exhibit many of the characteristic features of observed financial time… (More)

Necessary and sufficient conditions for the existence of a strictly stationary solution of the equations defining a general Lévy-driven continuous-parameter ARMA process with index set R are determined. Under these conditions the solution is shown to be unique and an explicit expression is given for the process as an integral with respect to the background… (More)

Necessary and sufficient conditions for the existence of a strictly stationary solution of the equations defining an ARMA(p, q) process driven by an independent and identically distributed noise sequence are determined. No moment assumptions on the driving noise sequence are made.

The deenition and properties of L evy-driven CARMA (continuous-time ARMA) processes are reviewed. Gaussian CARMA processes are special cases in which the driving L evy process is Brownian motion. The use of more general L evy processes permits the speciication of CARMA processes with a wide variety of marginal distributions which may be asymmetric and… (More)

The problem of tting continuous-time autoregressions (linear and non-linear) to closely and regularly spaced data is considered. For the linear case Jones (1981) and Bergstrom (1985) used state-space representations to compute exact maximum likelihood estimators and Phillips (1959) did so by tting an appropriate discrete-time ARMA process to the data. In… (More)

- Peter Brockwell
- 2004

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… (More)

Lattice algorithms for estimating the parameters of a multivariate autore-gression are generalized to deal with subset models in which some of the coefficient matrices are constrained to be zero. We first establish a recursive prediction-error version of the empirical Yule-Walker equations. The estimated coefficient matrices obtained from these recursions… (More)

Continuous-time autoregressive moving average (CARMA) processes with a non-negative kernel and driven by a non-decreasing Lévy process constitute a useful and very general class of stationary, non-negative continuous-time processes which have been used, in particular, for the modelling of stochastic volatility. Brockwell, Davis and Yang (2011) considered… (More)

We obtain necessary and sufficient conditions for the existence of strictly stationary solutions of multivariate ARMA equations with independent and identically distributed driving noise. For general ARMA(p, q) equations these conditions are expressed in terms of the coefficient polynomials of the defining equations and moments of the driving noise… (More)