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# Long time Hurst regularity of fractional SDEs and their ergodic means

Abstract : The fractional Brownian motion can be considered as a Gaussian field indexed by $(t,H) \in {\mathbb{R}_{+}\times (0,1)}$. On compact time intervals, it is known to be almost surely jointly H\"older continuous in time and Lipschitz continuous in the Hurst parameter $H$. 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. We obtain that the solutions and their ergodic means are almost surely H\"older continuous in $H$, uniformly in time. 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.
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https://hal-centralesupelec.archives-ouvertes.fr/hal-03695595
Contributor : Alexandre Richard Connect in order to contact the contributor
Submitted on : Wednesday, June 15, 2022 - 8:07:24 AM
Last modification on : Saturday, June 25, 2022 - 3:26:36 AM

### Identifiers

• HAL Id : hal-03695595, version 1
• ARXIV : 2206.06648

### Citation

El Mehdi Haress, Alexandre Richard. Long time Hurst regularity of fractional SDEs and their ergodic means. 2022. ⟨hal-03695595⟩

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