Joint estimation for volatility and drift parameters of ergodic jump diffusion processes via contrast function

TL;DR

This paper introduces a new contrast-based estimator for jointly estimating the drift and volatility parameters of ergodic jump diffusion processes from high-frequency data, without requiring specific discretization conditions.

Contribution

It extends existing methods by allowing joint estimation of drift and volatility in jump diffusions with minimal discretization constraints, using practical approximation techniques.

Findings

Estimator is asymptotically Gaussian.

Estimation feasible under nΔ^k → 0 for any k > 0.

Extends results to jump processes with finite activity.

Abstract

In this paper we consider an ergodic diffusion process with jumps whose drift coefficient depends on $\mu$ and volatility coefficient depends on $\sigma$, two unknown parameters. 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 $\theta$ := ($\mu$, $\sigma$), based on a contrast function, which is asymptotically gaussian without requiring any conditions on the rate at which $\Delta$n $\rightarrow$ 0, assuming a finite jump activity. This extends earlier results where a condition on the step discretization was needed (see [13],[28]) or where only the estimation of the drift parameter was considered (see [2]). In general situations, our contrast function is not explicit and in practise one has to resort to some approximation. We propose explicit approximations of the contrast function, such that the 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 [17] in the case of continuous processes. Efficient drift estimation, efficient volatility estimation,ergodic properties, high frequency data, L{\'e}vy-driven SDE, thresholding methods.