Asymptotic properties of functionals of increments of a continuous semimartingale with stochastic sampling times

TL;DR

This paper investigates the asymptotic behavior of functionals of continuous semimartingale increments under general stochastic sampling schemes, providing laws of large numbers and central limit theorems relevant for high-frequency financial data analysis.

Contribution

It extends asymptotic results and the realized kernel estimator to cases with random, tick-by-tick sampling times in financial data.

Findings

Established law of large numbers for functionals under stochastic sampling

Proved central limit theorem for normalized functionals

Extended realized kernel estimator to random sampling times

Abstract

This paper is concerned with asymptotic behavior of a variety of functionals of increments of continuous semimartingales. Sampling times are assumed to follow a rather general discretization scheme. If an underlying semimartingale is thought of as a financial asset price process, a general sampling scheme like the one employed in this paper is capable of reflecting what happens whenever the financial trading data are recorded in a tick-by-tick fashion. A law of large numbers and a central limit theorem are proved after an appropriate normalization. One application of our result is an extension of the realized kernel estimator of integrated volatility to the case of random sampling times.