Maximum likelihood estimation in the non-ergodic fractional Vasicek model

Stanislav Lohvinenko,Kostiantyn Ralchenko

doi:10.15559/19-vmsta140

Abstract

We investigate the fractional Vasicek model described by the stochastic differential equation $dX_t=(\alpha -\beta X_t)\,dt+\gamma \,dB^H_t$, $X_0=x_0$, driven by the fractional Brownian motion $B^H$ with the known Hurst parameter $H\in (1/2,1)$. We study the maximum likelihood estimators for unknown parameters $\alpha$ and $\beta$ in the non-ergodic case (when $\beta <0$) for arbitrary $x_0\in \mathbb{R}$, generalizing the result of Tanaka, Xiao and Yu (2019) for particular $x_0=\alpha /\beta$, derive their asymptotic distributions and prove their asymptotic independence.

Highlights

The present paper deals with the fractional Vasicek model of the form dXt = (α − βXt )dt + γ dBtH, X0 = x0 ∈ R, (1)where BH is the fractional Brownian motion with the Hurst index H ∈ (1/2, 1)
Where BH is the fractional Brownian motion with the Hurst index H ∈ (1/2, 1). It is a generalization of the classical interest rate model proposed by O
In order to use this model in practice, a theory of parameter estimation is necessary

Summary

Introduction

The least squares and ergodic-type estimators of unknown parameters α and β were studied in [27, 38, 39]. The corresponding MLEs of α and β were presented in [25] Their consistency and asymptotic normality were proved there for the case β > 0. More general results were proved in [26], where joint asymptotic normality of MLE of the vector parameter (α, β) was established. Tanaka et al [33] investigated asymptotic behavior of MLEs in the cases β = 0 and β < 0. The asymptotic behavior of the process X and of the estimators substantially depends on the sign of the parameter β. The asymptotic behavior of the MLEs in this case requires a separate study. Some auxiliary facts and results concerning modified Bessel functions of the first kind and MGFs related to the normal distribution are collected in the appendices

Preliminaries

Auxiliary results

Main results

A Modified Bessel function of the first kind

B MGFs related to the bivariate normal distribution