Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Stochastic differential equations driven by fractional Brownian motion: dependence on the Hurst parameter

Anna P. Kwossek, Andreas Neuenkirch, David J. Prömel

TL;DR

This survey analyzes how stochastic equations driven by fractional Brownian motion depend on the Hurst parameter $H\in(0,1)$. It links the Mandelbrot--van Ness representation to pathwise integration methods (Young, rough paths, and Doss--Sussmann) and characterizes $H$-dependent behavior for additive and non-additive noise, including ergodic and multi-dimensional settings. The results show Lipschitz-type stability of finite-horizon solutions and stationary laws with respect to $H$, while highlighting open problems for $H\le 1/2$ in the multi-dimensional rough path regime. The discussion extends to stochastic calculus for fBm, continuity properties of integrals and SPDEs under varying $H$, and implications for statistical applications and long-time behavior. Overall, the paper provides a comprehensive map of how fractional noise with different regularities influences SDEs and related systems through rigorous dependence results and methodological tools.

Abstract

Stochastic models with fractional Brownian motion as source of randomness have become popular since the early 2000s. Fractional Brownian motion (fBm) is a Gaussian process, whose covariance depends on the so-called Hurst parameter $H\in (0,1)$. Consequently, stochastic models with fBm also depend on the Hurst parameter $H$, and the stability of these models with respect to $H$ is an interesting and important question. In recent years, the continuous (or even smoother) dependence on the Hurst parameter has been studied for several stochastic models, including stochastic integrals with respect to fBm, stochastic differential equations (SDEs) driven by fBm and also stochastic partial differential equations with fractional noise, for different topologies, e.g., in law or almost surely, and for finite and infinite time horizons. In this manuscript, we give an overview of these results with a particular focus on SDE models.

Stochastic differential equations driven by fractional Brownian motion: dependence on the Hurst parameter

TL;DR

This survey analyzes how stochastic equations driven by fractional Brownian motion depend on the Hurst parameter . It links the Mandelbrot--van Ness representation to pathwise integration methods (Young, rough paths, and Doss--Sussmann) and characterizes -dependent behavior for additive and non-additive noise, including ergodic and multi-dimensional settings. The results show Lipschitz-type stability of finite-horizon solutions and stationary laws with respect to , while highlighting open problems for in the multi-dimensional rough path regime. The discussion extends to stochastic calculus for fBm, continuity properties of integrals and SPDEs under varying , and implications for statistical applications and long-time behavior. Overall, the paper provides a comprehensive map of how fractional noise with different regularities influences SDEs and related systems through rigorous dependence results and methodological tools.

Abstract

Stochastic models with fractional Brownian motion as source of randomness have become popular since the early 2000s. Fractional Brownian motion (fBm) is a Gaussian process, whose covariance depends on the so-called Hurst parameter . Consequently, stochastic models with fBm also depend on the Hurst parameter , and the stability of these models with respect to is an interesting and important question. In recent years, the continuous (or even smoother) dependence on the Hurst parameter has been studied for several stochastic models, including stochastic integrals with respect to fBm, stochastic differential equations (SDEs) driven by fBm and also stochastic partial differential equations with fractional noise, for different topologies, e.g., in law or almost surely, and for finite and infinite time horizons. In this manuscript, we give an overview of these results with a particular focus on SDE models.

Paper Structure

This paper contains 11 sections, 17 theorems, 110 equations.

Table of Contents

  1. Introduction
  2. SDEs driven by fBm
  3. Setup and standard definitions
  4. The Mandelbrot--van Ness representation of fBm and its dependence on the Hurst parameter
  5. The dependence of SDEs driven by fBm on the Hurst parameter
  6. SDEs with additive noise
  7. Ergodic SDEs with additive noise
  8. The Doss--Sussmann approach
  9. The multi-dimensional case for $H>1/2$
  10. The multi-dimensional case for $H\leq 1/2$
  11. Further results

Key Result

Theorem 2.2

Let $k \in \mathbb{N}$ and $T >0$. Then, there exists a process $B^{H,k} = (B_t^{H,k})_{t \in [0,T]}$ such that:

Theorems & Definitions (22)

  • Remark 2.1
  • Theorem 2.2
  • Proposition 2.3
  • Theorem 2.4
  • Remark 2.5
  • Lemma 3.1
  • proof
  • Proposition 3.2
  • Proposition 3.3
  • Theorem 3.4
  • ...and 12 more