Asymptotic growth of trajectories of multifractional Brownian motion, with statistical applications to drift parameter estimation

TL;DR

This paper develops a least-square estimator for the drift parameter in a multifractional Ornstein-Uhlenbeck model, proving its strong consistency and analyzing the asymptotic growth of multifractional Brownian motion trajectories.

Contribution

It introduces a new estimator for the drift parameter in multifractional models and establishes its strong consistency in non-ergodic cases, supported by asymptotic growth bounds.

Findings

Estimator is strongly consistent in non-ergodic cases.

Provides asymptotic bounds for the growth of multifractional Brownian motion trajectories.

Extends asymptotic growth analysis to general Gaussian processes.

Abstract

We construct the least-square estimator for the unknown drift parameter in the multifractional Ornstein-Uhlenbeck model and establish its strong consistency in the non-ergodic case. The proofs are based on the asymptotic bounds with probability 1 for the rate of the growth of the trajectories of multifractional Brownian motion (mBm) and of some other functionals of mBm, including increments and fractional derivatives. As the auxiliary results having independent interest, we produce the asymptotic bounds with probability 1 for the rate of the growth of the trajectories of the general Gaussian process and some functionals of it, in terms of the covariance function of its increments.