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Long time Hurst regularity of fractional SDEs and their ergodic means

El Mehdi Haress, Alexandre Richard

TL;DR

The paper addresses the problem of quantifying how fractional Brownian motion and additive fractional SDEs depend on the Hurst parameter over unbounded time and how ergodic means inherit this regularity. It develops variance bounds for simple and rectangular increments, and leverages a multiparameter Garsia-Rodemich-Rumsey framework along with a Gaussian-product identity to obtain almost-sure Hölder continuity in $H$, uniformly in time. The results show that solutions $Y_t^H$ and their ergodic means are Hölder in $H$ (and the invariant measure $\mu_H$ is Hölder in $H$ as well), with corresponding discrete-time analogues via Euler schemes. These findings underpin ergodic estimators of $H$ from discrete observations (HRstat) and provide tools for sensitivity analysis of SDEs driven by fBm with respect to the Hurst parameter.

Abstract

The fractional Brownian motion can be considered as a Gaussian field indexed by $(t,H)\in {\mathbb{R}_{+}\times (0,1)}$, where $H$ is the Hurst parameter. On compact time intervals, it is known to be almost surely jointly Hölder continuous in time and Lipschitz continuous in $H$. First, we extend this result to the whole time interval $\mathbb{R}_{+}$ and consider both simple and rectangular increments. Then we consider SDEs driven by fractional Brownian motion with contractive drift. The solutions and their ergodic means are proven to be almost surely Hölder continuous in $H$, uniformly in time. This result is used in a separate work for statistical applications. We also deduce a sensibility result of the invariant measure in $H$. The proofs are based on variance estimates of the increments of the fractional Brownian motion and fractional Ornstein-Uhlenbeck processes, multiparameter versions of the Garsia-Rodemich-Rumsey lemma and a combinatorial argument to estimate the expectation of a product of Gaussian variables.

Long time Hurst regularity of fractional SDEs and their ergodic means

TL;DR

The paper addresses the problem of quantifying how fractional Brownian motion and additive fractional SDEs depend on the Hurst parameter over unbounded time and how ergodic means inherit this regularity. It develops variance bounds for simple and rectangular increments, and leverages a multiparameter Garsia-Rodemich-Rumsey framework along with a Gaussian-product identity to obtain almost-sure Hölder continuity in , uniformly in time. The results show that solutions and their ergodic means are Hölder in (and the invariant measure is Hölder in as well), with corresponding discrete-time analogues via Euler schemes. These findings underpin ergodic estimators of from discrete observations (HRstat) and provide tools for sensitivity analysis of SDEs driven by fBm with respect to the Hurst parameter.

Abstract

The fractional Brownian motion can be considered as a Gaussian field indexed by , where is the Hurst parameter. On compact time intervals, it is known to be almost surely jointly Hölder continuous in time and Lipschitz continuous in . First, we extend this result to the whole time interval and consider both simple and rectangular increments. Then we consider SDEs driven by fractional Brownian motion with contractive drift. The solutions and their ergodic means are proven to be almost surely Hölder continuous in , uniformly in time. This result is used in a separate work for statistical applications. We also deduce a sensibility result of the invariant measure in . The proofs are based on variance estimates of the increments of the fractional Brownian motion and fractional Ornstein-Uhlenbeck processes, multiparameter versions of the Garsia-Rodemich-Rumsey lemma and a combinatorial argument to estimate the expectation of a product of Gaussian variables.
Paper Structure (39 sections, 26 theorems, 272 equations)

This paper contains 39 sections, 26 theorems, 272 equations.

Table of Contents

  1. Introduction
  2. Statistical applications.
  3. Organisation of the paper.
  4. Notations.
  5. Bounds on the variance of the increments of $B$ and $\mathbb{B}$
  6. Simple increments
  7. Step 1: Uniform upper bound on $\int_0^{\infty} \left( \partial_H K_1(x_h,t,s)\right)^2 \mathop{}\!\mathrm{d} s$.
  8. Step 2: Uniform upper bound on $\int_0^{t} \left( \partial_H K_2(x_h,t,s)\right)^2 \mathop{}\!\mathrm{d} s$.
  9. Step 3: Conclusion.
  10. Rectangular increments
  11. Almost sure results on the whole half-line for the fBm
  12. Statement of the results
  13. Garsia-Rodemich-Rumsey's lemma
  14. Proof of Theorem \ref{['thm:whole-regularity']}
  15. Bound on $I_1$ and $I_{3}$.
  16. ...and 24 more sections

Key Result

Proposition 2.1

Let $\mathcal{H}$ be a compact subset of $(0,1)$. There exists a constant $C$ such that for all $t'\ge t \ge 0$ and $H,H' \in \mathcal{H}$:

Theorems & Definitions (53)

  • Proposition 2.1
  • proof
  • Proposition 2.2
  • proof
  • Proposition 2.3
  • Proposition 2.4
  • Theorem 3.1
  • Remark 3.2
  • Remark 3.2
  • Remark 3.3
  • ...and 43 more