Optimal estimation of the supremum and occupation times of a self-similar Lévy process

Abstract

In this paper we present new theoretical results on optimal estimation of certain random quantities based on high frequency observations of a Lévy process. More specifically, we investigate the asymptotic theory for the conditional mean and conditional median estimators of the supremum/infimum of a linear Brownian motion and a strictly stable Lévy process. Another contribution of our article is the conditional mean estimation of the local time and the occupation time of a linear Brownian motion. We demonstrate that the new estimators are considerably more efficient compared to the classical estimators studied in e.g. [6,14,29,30,38]. Furthermore, we discuss pre-estimation of the parameters of the underlying models, which is required for practical implementation of the proposed statistics.

Discretization of the Lamperti representation of a positive self-similar Markov process

“…for all bounded continuous functions f and all bounded F -measurable Y , see also (16) below for further intuition. The standard example concerns Z being independent of F -measurable Y , and then the term mixing convergence is sometimes used.…”

“…Thus it is sufficient to take σ(ξ)-measurable Y in (4) to ensure the stable convergence, see also [18, p. 110]. Furthermore, it is sufficient to show (16) (Z n , ξ t1 , . .…”

“…Proof. It is sufficient to consider the process ξ (n) on some time interval [0, T ] jointly with ξ at some times t 1 < • • • < t k , see (16). That is, we need to show that ( ξ…”

“…Importantly, X is a strictly positive pssMp with Hurst parameter 1/α. Such processes naturally arise in limit theory concerned with extremes, first passage times and Skorokhod reflection, see [15,16] and references therein. Importantly, in case (I) the process X is Bessel-3.…”

“…Positive self-similar Markov processes (pssMp) have received a lot of attention in recent years, see [4,11] and a survey [23]. One class of examples is given by self-similar Lévy processes 'conditioned' to stay positive, which arise in various limit theorems concerned with extremes, first passage times and Skorokhod reflection [3,15,16]. Recall that X = (X t ) t≥0 is a pssMp if it is a positive strong Markov process with the self-similarity property: (X tc ) t≥0 with X 0 = x > 0 has the law of (c 1/α X t ) t≥0 with X 0 = c −1/α x for any c > 0, where 1/α > 0 is sometimes called the Hurst index.…”