Multi-Power Inference Beyond Semimartingales: Rough Gaussian Activity and Jumps

We study multiscale inference based on realized power variations when the continuous component of an observed process is rougher than a Brownian semimartingale. The continuous component is locally Gaussian with scaling exponent H < 1/2, and the model may additionally contain discontinuous activity. Realized p-variation then scales as ∆ pH-1 n , generating distinct regimes according to whether pH is below, at, or above one. At the critical boundary pH = 1, a continuous rough process may share the same first-order two-scale activity signature as a process containing jumps, creating an identification problem outside the usual semimartingale framework. We develop joint inference across several powers rather than relying on a single activity statistic. For the fractional Brownian benchmark, we derive a joint central limit theorem for the multi-power signatures and a roughness estimator computed from the same sample. This leads to a feasible quadratic specification statistic Q n whose null limit is chi-square. Under n -1/2 -local departures from the rough scaling relation, Q n has a noncentral chi-square limit, yielding an explicit asymptotic power function, while fixed identifiable alternatives are detected consistently. Gaussian Volterra extensions are obtained under explicit local-scaling and correlation-summability conditions. Monte Carlo experiments document the finite-sample size-power trade-off associated with using high powers and examine robustness to infinite-activity jumps, microstructure noise, and irregular sampling.