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

Fabienne Comte, Nicolas Marie

TL;DR

The paper develops nonparametric drift estimators for fractional SDEs driven by fractional Brownian motion using i.i.d. continuous paths. It replaces Itô-based regression with Skorokhod-based projection estimators and introduces a computable approximation $\widehat{b}_{m,\varepsilon}$ derived from two nearby initial conditions, plus an estimator for the derivative $b'$ and a computable primitive under restrictive conditions. The authors establish risk bounds decomposed into bias, variance, and negligible terms, and derive convergence rates in Fourier-Sobolev (trigonometric) and Hermite bases, highlighting a distinct impact of the Hurst parameter $H$ on variance terms. They also discuss an alternative derivative-driven estimator and provide detailed proofs for the concentration, density, and risk results, offering practical guidance for implementing nonparametric drift estimation in fractional diffusion models with functional data.

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 $\widehat b$ 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 $\widehat b$. Another bound is directly established on an estimator of $b'$ 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.

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

TL;DR

The paper develops nonparametric drift estimators for fractional SDEs driven by fractional Brownian motion using i.i.d. continuous paths. It replaces Itô-based regression with Skorokhod-based projection estimators and introduces a computable approximation derived from two nearby initial conditions, plus an estimator for the derivative and a computable primitive under restrictive conditions. The authors establish risk bounds decomposed into bias, variance, and negligible terms, and derive convergence rates in Fourier-Sobolev (trigonometric) and Hermite bases, highlighting a distinct impact of the Hurst parameter on variance terms. They also discuss an alternative derivative-driven estimator and provide detailed proofs for the concentration, density, and risk results, offering practical guidance for implementing nonparametric drift estimation in fractional diffusion models with functional data.

Abstract

This paper deals with nonparametric estimators of the drift function 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 of . 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 . Another bound is directly established on an estimator of for comparison. The consistency and rates of convergence are established for these estimators in the case of the compactly supported trigonometric basis or the -supported Hermite basis.

Paper Structure

This paper contains 30 sections, 17 theorems, 206 equations.

Table of Contents

  1. Introduction
  2. Stochastic integrals with respect to the fractional Brownian motion
  3. The pathwise stochastic integral
  4. Skorokhod's integral and density of the solution
  5. Projection estimator of the drift function
  6. A Skorokhod's integral based oracle
  7. The objective function
  8. The oracle
  9. Risk bounds on the oracle
  10. Rates in some usual bases
  11. Rates on Fourier-Sobolev spaces for the trigonometric basis
  12. Discussion on the Hermite example
  13. An approximate estimator
  14. Application context
  15. Risk bounds on the approximate estimator
  16. ...and 15 more sections

Key Result

Theorem 2.2

Let $x$ (resp. $w$) be a $\alpha$-Hölder (resp. $\beta$-Hölder) continuous map from $[0,T]$ into $\mathbb R$ with $\alpha,\beta\in ]0,1]$ such that $\alpha +\beta > 1$. There exists a unique continuous map $J_{x,w} : [0,T]\rightarrow\mathbb R$ such that for every $s,t\in [0,T]$ satisfying $s < t$ an The map $J_{x,w}$ is the Young integral of $x$ with respect to $w$ and $J_{x,w}(t) - J_{x,w}(s)$ is

Theorems & Definitions (24)

  • Definition 2.1
  • Theorem 2.2
  • Definition 2.3
  • Proposition 2.4
  • Definition 2.5
  • Proposition 2.6
  • Corollary 2.8
  • Theorem 2.9
  • Remark 2.10
  • Proposition 2.11
  • ...and 14 more