Peter J. Brockwell

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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)
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)
Continuous-time autoregressivemoving average (CARMA) processeswith a nonnegative kernel and driven by a nondecreasing Lévy process constitute a very general class of stationary, nonnegative continuous-time processes. In financial econometrics a stationary Ornstein–Uhlenbeck (or CAR(1)) process, driven by a nondecreasing Lévy process, was introduced by(More)
We consider the parametric estimation of the driving Lévy process of a multivariate continuous-time autoregressive moving average (MCARMA) process, which is observed on the discrete time grid (0, h, 2h, . . .). Beginning with a new state space representation, we develop a method to recover the driving Lévy process exactly from a continuous record of the(More)
Lattice algorithms for estimating the parameters of a multivariate autoregression 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)