The Annals of Statistics

Realized Laplace transforms for pure-jump semimartingales

Viktor Todorov and George Tauchen

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We consider specification and inference for the stochastic scale of discretely-observed pure-jump semimartingales with locally stable Lévy densities in the setting where both the time span of the data set increases, and the mesh of the observation grid decreases. The estimation is based on constructing a nonparametric estimate for the empirical Laplace transform of the stochastic scale over a given interval of time by aggregating high-frequency increments of the observed process on that time interval into a statistic we call realized Laplace transform. The realized Laplace transform depends on the activity of the driving pure-jump martingale, and we consider both cases when the latter is known or has to be inferred from the data.

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Ann. Statist. Volume 40, Number 2 (2012), 1233-1262.

First available in Project Euclid: 18 July 2012

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Zentralblatt MATH identifier

Primary: 62F12: Asymptotic properties of estimators 62M05: Markov processes: estimation
Secondary: 60H10: Stochastic ordinary differential equations [See also 34F05] 60J75: Jump processes

Laplace transform time-varying scale high-frequency data jumps inference


Todorov, Viktor; Tauchen, George. Realized Laplace transforms for pure-jump semimartingales. Ann. Statist. 40 (2012), no. 2, 1233--1262. doi:10.1214/12-AOS1006.

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Supplemental materials

  • Supplementary material: Supplement to “Realized Laplace transforms for pure-jump semimartingales”. This supplement contains proofs of the preliminary results in Section 7.1 as well as the proofs of Theorem 2, 4 and 5.