Is Brownian motion necessary to model high-frequency data?

TL;DR

This paper develops tests to determine whether high-frequency financial data require a continuous Brownian component or can be modeled solely with jumps, with applications indicating the necessity of including continuous parts.

Contribution

It introduces two novel statistical tests for identifying the presence or absence of a continuous component in high-frequency data models.

Findings

Both tests suggest the need for a continuous component in stock data models.

The tests effectively distinguish between pure jump and mixed processes.

Application to real data supports including Brownian motion in modeling.

Abstract

This paper considers the problem of testing for the presence of a continuous part in a semimartingale sampled at high frequency. We provide two tests, one where the null hypothesis is that a continuous component is present, the other where the continuous component is absent, and the model is then driven by a pure jump process. When applied to high-frequency individual stock data, both tests point toward the need to include a continuous component in the model.