Is it Brownian or fractional Brownian motion?

•We propose test statistics for testing Brownian motion against fractional Brownian motion.•Our test framework is robust to finite large jumps.•We extend bi-power variation to the inference of Hurst index.

Fractional Brownian motion embeds Brownian motion as a special case and offers more flexible diffusion component for pricing models. We propose test statistics based on bi-power variation for testing Brownian motion against fractional Brownian motion alternatives. To filter out the prevalent existence of finite large jumps, a truncation method based on Hurst index estimator is proposed. Simulation results confirm the consistency of jump truncation framework with desirable empirical size and viable empirical power for our tests.