On a Calculable Skorokhod's Integral Based Projection Estimator of the Drift Function in Fractional SDE

TL;DR

This paper introduces a new calculable projection estimator for the drift function in fractional SDEs driven by fractional Brownian motion, providing an $ ext{L}^2$-error bound for practical computation.

Contribution

It develops a Skorokhod's integral based estimator for the drift in fractional SDEs and establishes an $ ext{L}^2$-error bound for its approximation, enabling practical computation.

Findings

Provides an explicit $ ext{L}^2$-error bound for the estimator.

Demonstrates the estimator's calculability from observed data.

Extends drift estimation methods to fractional Brownian motion driven SDEs.

Abstract

This paper deals with a Skorokhod's integral based projection type estimator $\widehat b_m$ of the drift function $b_0$ computed from $N\in\mathbb N^*$ independent copies $X^1,\dots,X^N$ of the solution $X$ of $dX_t = b_0(X_t)dt +\sigma dB_t$, where $B$ is a fractional Brownian motion of Hurst index $H\in (1/2,1)$. Skorokhod's integral based estimators cannot be calculated directly from $X^1,\dots,X^N$, but in this paper an $\mathbb L^2$-error bound is established on a calculable approximation of $\widehat b_m$.