Nonparametric Estimation for I.I.D. Paths of Fractional SDE

Abstract

This paper deals with nonparametric estimators of the drift function $b$ computed from independent continuous observations, on a compact time interval, of the solution of a stochastic differential equation driven by the fractional Brownian motion (fSDE). First, a risk bound is established on a Skorokhod's integral based least squares oracle $\hat{b}$ of $b$. Thanks to the relationship between the solution of the fSDE and its derivative with respect to the initial condition, a risk bound is deduced on a calculable approximation of $\hat{b}$. Another bound is directly established on an estimator of $b^{\prime }$ for comparison. The consistency and rates of convergence are established for these estimators in the case of the compactly supported trigonometric basis or the $\mathbb{R}$-supported Hermite basis.

30 pages

References in corpus (2)

• Statistical aspects of the fractional stochastic calculus
• Almost Periodic and Periodic Solutions of Differential Equations Driven by the Fractional Brownian Motion with Statistical Application

Cited by in corpus (4)

• Nonparametric Estimation for I.I.D. Paths of a Martingale Driven Model with Application to Non-Autonomous Financial Models
• Almost Periodic and Periodic Solutions of Differential Equations Driven by the Fractional Brownian Motion with Statistical Application
• Nonparametric Drift Estimation from Diffusions with Correlated Brownian Motions
• On a Computable Skorokhod's Integral Based Estimator of the Drift Parameter in Fractional SDE