Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

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

Nicolas Marie

TL;DR

This work develops a Skorokhod-integral based least-squares estimator for the drift parameter in fractional SDEs driven by fractional Brownian motion with H ∈ (1/3,1). It shows computability challenges when H ≠ 1/2 and introduces a computable fixed-point estimator ˜θ_N for the H > 1/2 regime, with consistency and asymptotic confidence intervals, while deriving analogous results for H = 1/2 and outlining extensions to H < 1/2 via rough path methods. The paper provides nonasymptotic risk bounds that account for possible dependence among copies X^i, along with discrete-time approximations and numerical experiments validating the computable estimator. Key technical tools include Malliavin calculus, Skorokhod integration, and rough path/Young integration techniques. The results have practical implications for estimating drift in models driven by long-range dependent noise, with potential extensions to nonparametric projection estimators and broader fractional dynamics.

Abstract

This paper deals with a Skorokhod's integral based least squares type estimator $\widehatθ_N$ of the drift parameter $θ_0$ computed from $N\in\mathbb N^*$ (possibly dependent) copies $X^1,\dots,X^N$ of the solution $X$ of $dX_t =θ_0b(X_t)dt +σdB_t$, 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 $\widehatθ_N$ when $H = 1/2$. On the other hand, when $H\neq 1/2$, Skorokhod's integral based estimators as $\widehatθ_N$ cannot be computed from data, but in this paper some convergence results are established on a computable approximation of $\widehatθ_N$.

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

TL;DR

This work develops a Skorokhod-integral based least-squares estimator for the drift parameter in fractional SDEs driven by fractional Brownian motion with H ∈ (1/3,1). It shows computability challenges when H ≠ 1/2 and introduces a computable fixed-point estimator ˜θ_N for the H > 1/2 regime, with consistency and asymptotic confidence intervals, while deriving analogous results for H = 1/2 and outlining extensions to H < 1/2 via rough path methods. The paper provides nonasymptotic risk bounds that account for possible dependence among copies X^i, along with discrete-time approximations and numerical experiments validating the computable estimator. Key technical tools include Malliavin calculus, Skorokhod integration, and rough path/Young integration techniques. The results have practical implications for estimating drift in models driven by long-range dependent noise, with potential extensions to nonparametric projection estimators and broader fractional dynamics.

Abstract

This paper deals with a Skorokhod's integral based least squares type estimator of the drift parameter computed from (possibly dependent) copies of the solution of , where is a fractional Brownian motion of Hurst index . On the one hand, some convergence results are established on when . On the other hand, when , Skorokhod's integral based estimators as cannot be computed from data, but in this paper some convergence results are established on a computable approximation of .
Paper Structure (9 sections, 12 theorems, 182 equations, 3 figures, 1 table)

This paper contains 9 sections, 12 theorems, 182 equations, 3 figures, 1 table.

Table of Contents

  1. Introduction
  2. Case $H = 1/2$
  3. Case $H > 1/2$
  4. Basics on the Skorokhod integral
  5. Risk bound on the auxiliary estimator $\widehat{\theta}_{N}^{\mathfrak d}$
  6. A computable estimator
  7. A few elements in the case $H < 1/2$
  8. Numerical experiments
  9. Conclusion and perspectives

Key Result

Proposition 2.2

If $\mathcal{R}_N =\emptyset$, then for every $\alpha\in (0,1)$, where $u_. :=\phi^{-1}(.)$, $\phi$ is the standard normal distribution function, and

Figures (3)

  • Figure 1: Plots of $N\mapsto\widetilde{\theta}_N$ (black line) and of the bounds of the $95\%$-ACIs (dashed black lines) for Model 1 with $H = 0.7$ (left) and $H = 0.9$ (right).
  • Figure 2: Plots of $N\mapsto\widetilde{\theta}_N$ (black line) and the bounds of the $95\%$-ACIs (dashed black lines) for Model 2 with $H = 0.7$ (left) and $H = 0.9$ (right).
  • Figure 3: Plots of the mean error of the $\mathfrak d$-truncated estimator (black line) for Model 1 (left) and Model 2 (right).

Theorems & Definitions (29)

  • Remark 2.1
  • Proposition 2.2
  • proof
  • Remark 2.3
  • Proposition 2.4
  • proof
  • Remark 2.5
  • Remark 2.6
  • Proposition 2.7
  • proof
  • ...and 19 more