Abstract

Statistical inference for stochastic processes under high-frequency observations has been an active research area in econometrics and financial statistics for several decades. In this dissertation, we study volatility estimation for Itô semimartingales in the presence of highly active Lévy-type jumps. This problem is of fundamental importance in derivative pricing, risk management, portfolio allocation, and high-frequency financial econometrics. The central difficulty is that, when jumps have infinite activity and unbounded variation, the aggregate contribution of small jumps may remain non-negligible at the central limit scale and therefore distort inference for the continuous volatility component. The main techniques developed in this dissertation are based on truncation, smooth truncation, kernel localization, and recursive debiasing procedures designed to recover rate and variance efficiency beyond the classical bounded-variation jump regime. First, we study spot volatility estimation at a fixed time point. We consider a truncated kernel estimator for the local variance process and analyze its behavior under stable-like jumps of unbounded variation. By developing higher-order expansions for localized truncated moments, we identify the leading jump-induced bias terms that remain after truncation. These expansions show that the uncorrected truncated kernel estimator already attains the optimal n^1/4 convergence rate for a wider range of jump activity than previously known under comparable asymmetric settings. We then construct recursively debiased truncated kernel estimators and establish central limit theorems showing that the resulting estimators achieve rate-optimal and variance-efficient inference over substantially broader activity regimes, without imposing symmetry assumptions on the jump component as in earlier works in the litureature. Secondly, we study integrated volatility estimation, again, under highly active jumps. Instead of using ordinary hard truncation, we introduce a smooth analog of truncated realized quadratic variation. This modification allows the bias of the estimator to be automatically analyzed through small-time expansions based on the infinitesimal generator of the underlying process and Fourier-based methods. By carrying these expansions to arbitrary order, we construct closed-form recursive debiasing procedures that remove higher-order jump-induced bias terms systematically. The resulting estimators are explicit, avoid nonlinear estimating equations and numerical root-finding procedures as proposed in early works, and achieve both rate and variance efficiency, in the Cramér–Rao lower bound sense, throughout the entire stable-like infinite-variation regime 1 < Y < 2. Finally, we conduct extensive Monte Carlo experiments to evaluate the finite-sample perfor- mance of the proposed estimators for both spot and integrated volatility. The simulations examine different jump activity levels, sampling frequencies, truncation thresholds, and kernel choices. The results support the theoretical findings and show that the proposed debiasing procedures can substantially reduce bias and improve estimation accuracy in high-activity jump settings. In particular, the gains are most pronounced when the jump activity index is large, where classical truncation-based estimators and existing efficient alternatives may suffer from significant finite-sample deterioration.

Committee Chair

José Figueroa-López

Committee Members

Cooper Boniece; Jimin Ding; Likai Chen; Robert Lunde

Degree

Doctor of Philosophy (PhD)

Author's Department

Statistics

Author's School

Graduate School of Arts and Sciences

Document Type

Dissertation

Date of Award

8-11-2026

Language

English (en)

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Available for download on Monday, August 07, 2028

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