Drift parameter estimation for nonlinear stochastic differential equations driven by fractional Brownian motion

TL;DR

This paper establishes the strong consistency of the least squares estimator for the drift coefficient in nonlinear fractional stochastic differential equations driven by fractional Brownian motion, using ergodic theory and Malliavin calculus.

Contribution

It introduces a novel proof of estimator consistency for systems with one-sided dissipative Lipschitz drift and fractional noise with Hurst parameter in (1/4, 1).

Findings

Proves strong consistency of the least squares estimator.

Derives a maximum inequality for Skorohod integrals.

Provides tools for parameter estimation in fractional SDEs.

Abstract

We derive the strong consistency of the least squares estimator for the drift coefficient of a fractional stochastic differential system. The drift coeffcient is one-sided dissipative Lipschitz and the driving noise is additive and fractional with Hurst parameter $H \in (\frac{1}{4}, 1)$. We assume that continuous observation is possible. The main tools are ergodic theorem and Malliavin calculus. As a by-product, we derive a maximum inequality for Skorohod integrals, which plays an important role to obtain the strong consistency of the least squares estimator.