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Volatility estimators for discretely sampled Lévy processes. (English) Zbl 1114.62109
Summary: This paper studies the estimation of the volatility parameter in a model where the driving process is a Brownian motion or a more general symmetric stable process that is perturbed by another Lévy process. We distinguish between a parametric case, where the law of the perturbing process is known, and a semiparametric case, where it is not. In the parametric case, we construct estimators which are asymptotically efficient. In the semiparametric case, we can obtain asymptotically efficient estimators by sampling at a sufficiently high frequency, and these estimators are efficient uniformly in the law of the perturbing process.

MSC:
62P05 Applications of statistics to actuarial sciences and financial mathematics
62F12 Asymptotic properties of parametric estimators
62M05 Markov processes: estimation; hidden Markov models
60H30 Applications of stochastic analysis (to PDEs, etc.)
62E20 Asymptotic distribution theory in statistics
60J70 Applications of Brownian motions and diffusion theory (population genetics, absorption problems, etc.)
Software:
STABLE
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