Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

On a Calculable Skorokhod's Integral Based Projection Estimator of the Drift Function in Fractional SDE

Nicolas Marie

TL;DR

The paper tackles nonparametric drift estimation for a fractional SDE driven by a Brownian input with Hurst index $H\in(\tfrac12,1)$, using $N$ independent copies to form a projection-type Skorokhod-based estimator that is not directly computable. It introduces a calculable fixed-point surrogate $\widetilde b_m$ via a map $\Phi_m$ on a function class, and derives explicit $L^2$-error bounds for the auxiliary estimator and the fixed-point estimator under suitable density and dissipativity conditions. The main contributions are (i) an $L^2$-error bound for the calculable approximation of the Skorokhod-based projection, (ii) a contraction-based existence and uniqueness result for the fixed-point estimator, and (iii) a detailed analysis of the bias-variance tradeoffs and rate implications for choices of $m$ and the time horizon $T$, tying Malliavin calculus to practical projection-estimation in fractional SDEs. These results enable principled drift estimation from finite, independent data segments in fractional diffusion models, with explicit dependence on $N$, $T$, and $H$.

Abstract

This paper deals with a Skorokhod's integral based projection type estimator $\widehat b_m$ of the drift function $b_0$ computed from $N\in\mathbb N^*$ independent copies $X^1,\dots,X^N$ of the solution $X$ of $dX_t = b_0(X_t)dt +σdB_t$, where $B$ is a fractional Brownian motion of Hurst index $H\in (1/2,1)$. Skorokhod's integral based estimators cannot be calculated directly from $X^1,\dots,X^N$, but in this paper an $\mathbb L^2$-error bound is established on a calculable approximation of $\widehat b_m$.

On a Calculable Skorokhod's Integral Based Projection Estimator of the Drift Function in Fractional SDE

TL;DR

The paper tackles nonparametric drift estimation for a fractional SDE driven by a Brownian input with Hurst index , using independent copies to form a projection-type Skorokhod-based estimator that is not directly computable. It introduces a calculable fixed-point surrogate via a map on a function class, and derives explicit -error bounds for the auxiliary estimator and the fixed-point estimator under suitable density and dissipativity conditions. The main contributions are (i) an -error bound for the calculable approximation of the Skorokhod-based projection, (ii) a contraction-based existence and uniqueness result for the fixed-point estimator, and (iii) a detailed analysis of the bias-variance tradeoffs and rate implications for choices of and the time horizon , tying Malliavin calculus to practical projection-estimation in fractional SDEs. These results enable principled drift estimation from finite, independent data segments in fractional diffusion models, with explicit dependence on , , and .

Abstract

This paper deals with a Skorokhod's integral based projection type estimator of the drift function computed from independent copies of the solution of , where is a fractional Brownian motion of Hurst index . Skorokhod's integral based estimators cannot be calculated directly from , but in this paper an -error bound is established on a calculable approximation of .
Paper Structure (4 sections, 7 theorems, 79 equations)

This paper contains 4 sections, 7 theorems, 79 equations.

Table of Contents

  1. Introduction
  2. Basics on the Skorokhod integral
  3. An $\mathbb L^2$-error bound on the auxiliary estimator $\widehat{b}_m$
  4. Existence, uniqueness and $\mathbb L^2$-error bound on the fixed point estimator $\widetilde{b}_m$

Key Result

Proposition 2.2

The map $\mathbf D$ is closable from $\mathbb L^2(\Omega;\mathbb R)$ into $\mathbb L^2(\Omega;\mathcal{H})$. Its domain in $\mathbb L^2(\Omega;\mathbb R)$, denoted by $\mathbb D^{1,2}$, is the closure of the smooth functionals space for the norm $\|.\|_{1,2}$ defined by The Malliavin derivative of $F\in\mathbb D^{1,2}$ at time $s\in [0,T]$ is denoted by $\mathbf D_sF$.

Theorems & Definitions (17)

  • Definition 2.1
  • Proposition 2.2
  • Definition 2.3
  • Proposition 2.4
  • proof
  • Proposition 2.5
  • Example 3.2
  • Proposition 3.3
  • proof
  • Example 4.2
  • ...and 7 more