Statistical Estimations for Non-Ergodic Vasicek Model Driven by Two Types of Gaussian Processes

TL;DR

This paper investigates the asymptotic behavior of least squares estimators for parameters in non-ergodic Vasicek models driven by seven different Gaussian processes, providing new inner product formulas and conditions for their distributional properties.

Contribution

It introduces novel inner product formulas for Gaussian processes and establishes joint asymptotic distributions of estimators in non-ergodic Vasicek models driven by these processes.

Findings

Derived joint asymptotic distributions for estimators.

Developed new inner product formulas for Gaussian processes.

Validated conditions for distributional convergence.

Abstract

We study the joint asymptotic distribution of the least squares estimator of the parameter $(\theta,\,\mu)$ for the non-ergodic Vasicek models driven by seven specific Gaussian processes. %The similar result concerning to the non-ergodic Ornstein-Uhlenbeck process is a by-product. To facilitate the proofs, we extract two common hypotheses from the covariance functions of the seven Gaussian processes and develop two types of new inner product formulas for functions of bounded variation in the reproducing kernel Hilbert space of the Gaussian processes. The integration by parts for normalized bounded variation functions is essential to the inner product formulas. We apply the inner product formulas of the seven Gaussian processes to check the set of conditions of Es-Sebaiy, Es.Sebaiy (2021).