Estimation Of all parameters in the Fractional Ornstein-Uhlenbeck model under discrete observations

TL;DR

This paper develops consistent estimators for all parameters of a fractional Ornstein-Uhlenbeck process from discrete observations, allowing fixed step size and using ergodic theory and Malliavin calculus.

Contribution

It introduces novel ergodic-type estimators for all parameters in the fractional Ornstein-Uhlenbeck model that work with fixed observation step size.

Findings

Establishes strong consistency of the estimators.

Derives the rate of convergence for the estimators.

Allows fixed step size in observations, reflecting practical scenarios.

Abstract

Let the Ornstein-Uhlenbeck process $(X_t)_{t\ge0}$ driven by a fractional Brownian motion $B^{H }$, described by $dX_t = -\theta X_t dt + \sigma dB_t^{H }$ be observed at discrete time instants $t_k=kh$, $k=0, 1, 2, \cdots, 2n+2 $. We propose ergodic type statistical estimators $\hat \theta_n $, $\hat H_n $ and $\hat \sigma_n $ to estimate all the parameters $\theta $, $H $ and $\sigma $ in the above Ornstein-Uhlenbeck model simultaneously. We prove the strong consistence and the rate of convergence of the estimators. The step size $h$ can be arbitrarily fixed and will not be forced to go zero, which is usually a reality. The tools to use are the generalized moment approach (via ergodic theorem) and the Malliavin calculus.