On a Computable Skorokhod's Integral Based Estimator of the Drift Parameter in Fractional SDE

TL;DR

This paper introduces a new estimator for the drift parameter in fractional SDEs using Skorokhod's integral, analyzing its convergence properties for different Hurst indices and proposing a computable approximation when direct calculation is infeasible.

Contribution

It develops a Skorokhod integral-based estimator for fractional SDEs and establishes its convergence, including a practical approximation for cases where direct computation is not possible.

Findings

Convergence results for the estimator when H=1/2.

Convergence analysis of a computable approximation for H≠1/2.

The estimator's applicability to fractional Brownian motion-driven SDEs.

Abstract

This paper deals with a Skorokhod's integral based least squares type estimator $\widehat\theta_N$ of the drift parameter $\theta_0$ computed from $N\in\mathbb N^*$ (possibly dependent) copies $X^1,\dots,X^N$ of the solution $X$ of $dX_t =\theta_0b(X_t)dt +\sigma dB_t$, where $B$ is a fractional Brownian motion of Hurst index $H\in (1/3,1)$. On the one hand, some convergence results are established on $\widehat\theta_N$ when $H = 1/2$. On the other hand, when $H\neq 1/2$, Skorokhod's integral based estimators as $\widehat\theta_N$ cannot be computed from data, but in this paper some convergence results are established on a computable approximation of $\widehat\theta_N$.