LAN property for stochastic differential equations driven by fractional Brownian motion of Hurst parameter $H\in(1/4,1/2)$

TL;DR

This paper investigates the LAN property for stochastic differential equations driven by fractional Brownian motion with Hurst parameter between 1/4 and 1/2, deriving likelihood ratios and asymptotic normality results.

Contribution

It provides the first derivation of the likelihood ratio and LAN property for SDEs driven by fractional Brownian motion with H in (1/4, 1/2).

Findings

Likelihood ratio formula derived for H in (1/4, 1/2)

Proved local asymptotic normality for the model

Identified different convergence rates depending on parameter conditions

Abstract

In this paper, we consider the problem of estimating the drift parameter of solution to the stochastic differential equation driven by a fractional Brownian motion with Hurst parameter less than $1/2$ under complete observation. We derive a formula for the likelihood ratio and prove local asymptotic normality when $H \in (1/4,1/2)$. Our result shows that the convergence rate is $T^{-1/2}$ for the parameters satisfying a certain equation and $T^{-(1-H)}$ for the others.