Contrast function estimation for the drift parameter of ergodic jump diffusion process

TL;DR

This paper introduces an efficient estimator for the drift parameter of an ergodic jump diffusion process, applicable with high-frequency data and allowing for non-summable jumps without restrictive sampling rate conditions.

Contribution

It extends existing methods by enabling efficient drift estimation in jump diffusions with non-summable jumps and arbitrary sampling rates, broadening applicability.

Findings

Estimator is efficient without conditions on sampling rate.

Explicit approximations are provided for finite jump activity.

Method applies to processes with non-summable jumps.

Abstract

In this paper we consider an ergodic diffusion process with jumps whose drift coefficient depends on an unknown parameter $\theta$. We suppose that the process is discretely observed at the instants (t n i)i=0,...,n with $\Delta$n = sup i=0,...,n--1 (t n i+1 -- t n i) $\rightarrow$ 0. We introduce an estimator of $\theta$, based on a contrast function, which is efficient without requiring any conditions on the rate at which $\Delta$n $\rightarrow$ 0, and where we allow the observed process to have non summable jumps. This extends earlier results where the condition n$\Delta$ 3 n $\rightarrow$ 0 was needed (see [10],[24]) and where the process was supposed to have summable jumps. Moreover, in the case of a finite jump activity, we propose explicit approximations of the contrast function, such that the efficient estimation of $\theta$ is feasible under the condition that n$\Delta$ k n $\rightarrow$ 0 where k > 0 can be arbitrarily large. This extends the results obtained by Kessler [15] in the case of continuous processes. L{\'e}vy-driven SDE, efficient drift estimation, high frequency data, ergodic properties, thresholding methods.