Is the Driving Force of a Continuous Process a Brownian Motion or Fractional Brownian Motion?

Abstract

It? ’ s semimartingale driven by a Brownian motion is typically used in modeling the asset prices, interest rates and exchange rates, and so on. However, the assumption of Brownian motion as a driving force of the underlying asset price processes is rarely contested in practice. This naturally raises the question of whether this assumption is really appropriate. In the paper we propose a statistical test to answer the above question using high frequency data. The test can be used to validate the assumption of semimartingale framework and test for the existence of the long run dependence captured by the fractional Brownian motion in a parsimonious way. Asymptotic properties of the test statistics are investigated. Simulations justify the performance of the test. Real data sets are also analyzed.