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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)

- 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)

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)

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

Recently, there has been increasing interest in continuous-time stochastic models with jumps, a class of models which has applications in the fields of finance, insurance mathematics and storage theory, to name just a few. In this chapter we shall collect known results about a prominent class of these continuoustime models with jumps, namely the class of… (More)

- Peter J. Brockwell, Eckhard Schlemm
- J. Multivariate Analysis
- 2013

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)

- E Trucco, P J Brockwell
- Journal of theoretical biology
- 1968

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)