Statistical inference for Vasicek-type model driven by Hermite processes

TL;DR

This paper develops estimators for the drift parameters of a Vasicek-type model driven by Hermite processes, which include fractional Brownian motion and non-Gaussian processes, proving their consistency and asymptotic properties.

Contribution

It introduces parameter estimation methods for Vasicek models driven by Hermite processes, extending analysis to non-Gaussian, long-range dependent cases.

Findings

Estimates are strongly consistent for all H and q.

Asymptotic distributions are characterized for the estimators.

Method applies to both Gaussian and non-Gaussian Hermite-driven models.

Abstract

Let $Z$ denote a Hermite process of order $q \geq 1$ and self-similarity parameter $H \in (\frac{1}{2}, 1)$. This process is $H$-self-similar, has stationary increments and exhibits long-range dependence. When $q=1$, it corresponds to the fractional Brownian motion, whereas it is not Gaussian as soon as $q\geq 2$. In this paper, we deal with a Vasicek-type model driven by $Z$, of the form $dX_t = a(b - X_t)dt +dZ_t$. Here, $a > 0$ and $b \in \mathbb{R}$ are considered as unknown drift parameters. We provide estimators for $a$ and $b$ based on continuous-time observations. For all possible values of $H$ and $q$, we prove strong consistency and we analyze the asymptotic fluctuations.