Estimation for Lévy-driven CARMA processes
Date
2008
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Abstract
This thesis explores parameter estimation for Lévy-driven continuous-time autoregressive moving average (CARMA) processes, using uniformly and closely spaced discrete-time observations. Specifically, we focus on developing estimation techniques and asymptotic properties of the estimators for three particular families of Lévy-driven CARMA processes. Estimation for the first family, Gaussian autoregressive processes, was developed by deriving exact conditional maximum likelihood estimators of the parameters under the assumption that the process is observed continuously. The resulting estimates are expressed in terms of stochastic integrals which are then approximated using the available closely-spaced discrete-time observations. We apply the results to both linear and non-linear autoregressive processes. For the second family, non-negative Lévy-driven Ornestein-Uhlenbeck processes, we take advantage of the non-negativity of the increments of the driving Lévy process to derive a highly efficient estimation procedure for the autoregressive coefficient when observations are available at uniformly spaced times. Asymptotic properties of the estimator are also studied and a procedure for obtaining estimates of the increments of the driving Lévy process is developed. These estimated increments are important for identifying the nature of the driving Lévy process and for estimating its parameters. For the third family, non-negative Lévy-driven CARMA processes, we estimate the coefficients by maximizing the Gaussian likelihood of the observations and discuss the asymptotic properties of the estimators. We again show how to estimate the increments of the background driving Lévy process and hence to estimate the parameters of the Lévy process itself. We assess the performance of our estimation procedures by simulations and use them to fit models to real data sets in order to determine how the theory applies in practice.
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Subject
continuous-time ARMA
continuous-time autoregression
Lévy process
sampled process
stochastic differential equation
stochastic volatility
statistics