Statistical analysis for discretely observed Lévy processes
We consider Lévy processes X_t given by the Lévy triplet (mu(theta),sigma^2(theta),nu_theta(dx)), where mu(theta) denotes the drift, sigma^2(theta) the diffusion part, nu_theta(dx) the Lévy measure, and theta is some unknown parameter. Our aim is to establish efficiency results for the estimation of theta, when the process is observed at discrete time points. First, we derive two methods approximating the density or distribution function of X_t in terms of nu_theta and find integral relations when t tends to zero. Then, we prove local asymptotic normality under different sampling schemes and conditions on the Lévy measure, including stable, Gamma, Normal inverse Gaussian and generalized hyperbolic Lévy processes. Furthermore, we apply our results to martingale estimating functions to obtain efficient estimators.
Lokalasymptotische Normalität, diskrete Beobachtungen, Dichteapproximationen, Lévy processes, local asymptotic normality, discrete observations, estimating functions, density approximations
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