# High-frequency inference for stochastic processes with jumps of infinite activity

Mies, Fabian; Steland, Ansgar (Thesis advisor); Bibinger, Markus (Thesis advisor); Podolskij, Mark (Thesis advisor)

*Aachen (2020)* [Dissertation / PhD Thesis]

*Page(s): 1 Online-Ressource (117 Seiten) : Illustrationen*

Abstract

Jump-type discontinuities are an integral part of the well-established theory of stochastic processes, notably of the theory of semimartingales. In this thesis, we study statistical inference for processes with infinitely many jumps, i.e. the jumps have infinite activity. These processes may be conveniently characterized in terms of their jump-activity index, also known as the Blumenthal-Getoor (BG) index in the case of Lévy processes. The BG index compares the small jumps of a process to those of an alpha-stable Lévy process. We are concerned with estimation of this index itself, as well as with inference for the drift and volatility which is robust to infinitely active jumps. The estimators are based on discrete equidistant observations, and we consider the asymptotic setting of a shrinking grid size and a potentially increasing time span of observations. Statistical methods for this regime are referred to as high-frequency inference, and we present an extensive literature review of this field of study (chapter 2). The first contribution of this thesis concerns non-parametric inference for Markovian semimartingales (chapter 3). We study Nadaraya-Watson type estimators of the drift term and of the jump-activity index, and derive their asymptotic distribution in a joint high-frequency and long-time-span limit regime. The drift estimator is asymptotically normal even if the increments have infinite variance, and its asymptotic distribution is unaffected by the presence of the jump component. In contrast to most parts of the literature, we allow the jump-activity index to be state-dependent, and we show that existing estimators can be adapted to this non-parametric situation. The results rely on an analytical treatment of the infinitesimal generator of the Markov process, allowing us to derive approximations of conditional expectations with explicit error bounds and to conveniently control various bias terms. The second set of results is motivated by an open problem concerning the estimation of the jump-activity index in a pure high-frequency setting (chapter 4). In the basic case of Lévy processes with a non-vanishing diffusion component, all existing estimators for the BG index admit rates of convergence which are suboptimal by a small polynomial factor. After reviewing the existing approaches, we present a new estimator which achieves the optimal rate of convergence up to a logarithmic factor. It is also demonstrated that the Fisher matrix of a corresponding parametric sub-model is asymptotically singular. Hence, we conjecture that the rate achieved by our estimator is in fact optimal. The novel procedure is based on a single set of estimating equations, which also yields a new optimal estimator of the volatility parameter in the presence of jumps of infinite variation, as well as estimators for the so-called successive BG indices. Simulations suggest that the proposed estimators incur smaller errors in finite samples than the existing alternatives for the estimation of volatility as well as for the BG index.

### Identifier

- REPORT NUMBER: RWTH-2020-07022