## The Annals of Statistics

### Realized Laplace transforms for pure-jump semimartingales

#### Abstract

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.

#### Article information

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

Dates
First available in Project Euclid: 18 July 2012

https://projecteuclid.org/euclid.aos/1342625467

Digital Object Identifier
doi:10.1214/12-AOS1006

Mathematical Reviews number (MathSciNet)
MR2985949

Zentralblatt MATH identifier
1274.62191

#### Citation

Todorov, Viktor; Tauchen, George. Realized Laplace transforms for pure-jump semimartingales. Ann. Statist. 40 (2012), no. 2, 1233--1262. doi:10.1214/12-AOS1006. https://projecteuclid.org/euclid.aos/1342625467

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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.