Nonparametric tests for pathwise properties of semimartingales

TL;DR

This paper introduces two nonparametric tests to analyze the pathwise properties of semimartingales, focusing on detecting continuous components and jump variation types, with applications to financial data.

Contribution

It presents novel nonparametric tests for identifying continuous martingale parts and jump variation types in semimartingales, applicable to real financial data.

Findings

Detected non-zero Brownian component in exchange rate and futures data

Identified finite variation jumps in financial signals

Validated tests through simulation studies

Abstract

We propose two nonparametric tests for investigating the pathwise properties of a signal modeled as the sum of a L\'{e}vy process and a Brownian semimartingale. Using a nonparametric threshold estimator for the continuous component of the quadratic variation, we design a test for the presence of a continuous martingale component in the process and a test for establishing whether the jumps have finite or infinite variation, based on observations on a discrete-time grid. We evaluate the performance of our tests using simulations of various stochastic models and use the tests to investigate the fine structure of the DM/USD exchange rate fluctuations and SPX futures prices. In both cases, our tests reveal the presence of a non-zero Brownian component and a finite variation jump component.