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Nonparametric Estimation from Correlated Copies of a Drifted Process

Nicolas Marie

TL;DR

The paper addresses nonparametric estimation of a drift function $b_0$ and its derivative from multiple correlated copies of a drifted process using a pathwise Young-integral framework. It proposes simple estimators $\hat b=\bar X$ and $\hat b_m'$ based on projections onto a finite-dimensional basis, and proves nonasymptotic risk bounds that separate bias and variance components, with a Gaussian-case sharpening and adaptive model selection. The work covers constructions of correlated copies from linear (fractional) diffusions, interacting particle systems, and long-time fractional-noise observations, plus numerical experiments illustrating finite-sample performance. This framework enables robust drift and derivative estimation under dependence among copies, with practical relevance to finance, pharmacokinetics, and functional data analysis.

Abstract

This paper presents several situations leading to the observation of multiple correlated copies of a drifted process, and then non-asymptotic risk bounds are established on nonparametric estimators of the drift function $b_0$ and its derivative. For drifted Gaussian processes with a regular enough covariance function, a sharper risk bound is established on the estimator of $b_0'$, and a model selection procedure is provided with theoretical guarantees.

Nonparametric Estimation from Correlated Copies of a Drifted Process

TL;DR

The paper addresses nonparametric estimation of a drift function and its derivative from multiple correlated copies of a drifted process using a pathwise Young-integral framework. It proposes simple estimators and based on projections onto a finite-dimensional basis, and proves nonasymptotic risk bounds that separate bias and variance components, with a Gaussian-case sharpening and adaptive model selection. The work covers constructions of correlated copies from linear (fractional) diffusions, interacting particle systems, and long-time fractional-noise observations, plus numerical experiments illustrating finite-sample performance. This framework enables robust drift and derivative estimation under dependence among copies, with practical relevance to finance, pharmacokinetics, and functional data analysis.

Abstract

This paper presents several situations leading to the observation of multiple correlated copies of a drifted process, and then non-asymptotic risk bounds are established on nonparametric estimators of the drift function and its derivative. For drifted Gaussian processes with a regular enough covariance function, a sharper risk bound is established on the estimator of , and a model selection procedure is provided with theoretical guarantees.

Paper Structure

This paper contains 14 sections, 12 theorems, 137 equations, 3 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Preliminaries on the Young integral
  3. Some dynamics and observation schemes leading to Model (\ref{['main_model_copies']})
  4. From (fractional) diffusions and interacting particle systems to Model (\ref{['main_model_copies']})
  5. Copies of a linear (fractional) diffusion
  6. A basic interacting particle system
  7. From long-time observation to correlated copies
  8. Nonparametric estimators of $b_0$ and $b_0'$
  9. Nonparametric estimator of $b_0$
  10. Nonparametric estimator of $b_0'$
  11. Back to the models of Section \ref{['section_examples']}
  12. Numerical experiments
  13. Estimation in Models (\ref{['interacting_particle_system']}) and (\ref{['long_time_observation_copies_fractional_model']}) via $\widehat{b}$
  14. Estimation in Model (\ref{['linear_SDE_correlated_noises']}) via $\widehat{b}'$

Key Result

Theorem 2.2

Consider $\beta\in (0,1]$ such that $\alpha +\beta > 1$, and let $h$ (resp. $x$) be a $\beta$-Hölder (resp. $\alpha$-Hölder) continuous function from $[0,T]$ into $\mathbb R$. Then, there exists a unique $\alpha$-Hölder continuous function $J_{h,x} : [0,T]\rightarrow\mathbb R$ such that, for any $s, The Young integral on $[s,t]$ of $h$ with respect to $x$ is defined by

Figures (3)

  • Figure 1: Plots of $5$ estimations (dashed black) of the true function $\texttt{b}_0$ (red) in Model (\ref{['interacting_particle_system']}) ($N = 100$ copies).
  • Figure 2: Plots of 5 estimations (dashed black) of the true function $b_0$ (red) in Model (\ref{['long_time_observation_copies_fractional_model']}) for $\delta = 2$ and $H = 0.6,0.9$ ($N = 50$ copies).
  • Figure 3: Plots of 5 adaptive estimations (dashed black) of the true function $\texttt{b}_0$ (red) in Equation (\ref{['linear_SDE_correlated_noises']}) for $\gamma = 0,0.5,0.75$ ($N = 100$ copies).

Theorems & Definitions (28)

  • Definition 2.1
  • Theorem 2.2
  • Definition 2.3
  • Proposition 2.4
  • proof
  • Proposition 3.1
  • proof
  • Proposition 3.2
  • proof
  • Proposition 4.1
  • ...and 18 more