Realized Laplace transforms for pure-jump semimartingales

TL;DR

This paper introduces a nonparametric method called realized Laplace transform for estimating the stochastic scale of pure-jump semimartingales from high-frequency data, accommodating increasing data span and decreasing observation mesh.

Contribution

It develops a novel estimation technique based on realized Laplace transforms for pure-jump processes with locally stable Lévy densities, handling both known and unknown activity levels.

Findings

Consistent estimation of the stochastic scale using realized Laplace transforms.

Effective inference method for pure-jump semimartingales with high-frequency data.

Applicability to cases with unknown activity of the driving martingale.

Abstract

We consider specification and inference for the stochastic scale of discretely-observed pure-jump semimartingales with locally stable L\'{e}vy 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.