Multi-Power Inference Beyond Semimartingales: Rough Gaussian Activity and Jumps
Abstract
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 ΔnpH−1, 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 Qn whose null limit is chi-square. Under n−1/2-local departures from the rough scaling relation, Qn 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.
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BibTeX
@misc{kamdem2026multipower,
title = {{Multi-Power Inference Beyond Semimartingales: Rough Gaussian Activity and Jumps}},
author = {Jules Sadefo Kamdem},
howpublished = {HAL},
year = {2026},
month = may,
url = {https://hal.science/hal-05756191v1},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Multi-Power Inference Beyond Semimartingales: Rough Gaussian Activity and Jumps
AU - Jules Sadefo Kamdem
PY - 2026
DA - 2026/05/30
PB - HAL
UR - https://hal.science/hal-05756191v1
AB - 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 Δ_n^pH−1, 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.
ER -