Jump activity estimation for pure-jump semimartingales via self-normalized statistics

TL;DR

This paper introduces a new nonparametric estimator for the jump-activity index of pure-jump semimartingales, offering efficiency improvements and robustness over existing methods, especially in stochastic and jump-diffusion settings.

Contribution

The paper develops a self-normalized, characteristic function-based estimator for the jump-activity index that outperforms existing power variation methods in efficiency and applicability.

Findings

Estimator reduces asymptotic variance for higher eta values.

Constant asymptotic variance improves efficiency in stochastic settings.

Faster convergence rate for eta=2 (jump-diffusion case).

Abstract

We derive a nonparametric estimator of the jump-activity index $\beta$ of a "locally-stable" pure-jump It\^{o} semimartingale from discrete observations of the process on a fixed time interval with mesh of the observation grid shrinking to zero. The estimator is based on the empirical characteristic function of the increments of the process scaled by local power variations formed from blocks of increments spanning shrinking time intervals preceding the increments to be scaled. The scaling serves two purposes: (1) it controls for the time variation in the jump compensator around zero, and (2) it ensures self-normalization, that is, that the limit of the characteristic function-based estimator converges to a nondegenerate limit which depends only on $\beta$. The proposed estimator leads to nontrivial efficiency gains over existing estimators based on power variations. In the L\'{e}vy case, the asymptotic variance decreases multiple times for higher values of $\beta$. The limiting asymptotic variance of the proposed estimator, unlike that of the existing power variation based estimators, is constant. This leads to further efficiency gains in the case when the characteristics of the semimartingale are stochastic. Finally, in the limiting case of $\beta=2$, which corresponds to jump-diffusion, our estimator of $\beta$ can achieve a faster rate than existing estimators.