# Rate of convergence to equilibrium of fractional driven stochastic differential equations with rough multiplicative noise

Abstract : We investigate the problem of the rate of convergence to equilibrium for ergodic stochastic differential equations driven by fractional Brownian motion with Hurst parameter $H\in (1/3,1)$ and multiplicative noise component $\sigma$. When $\sigma$ is constant and for every $H\in (0,1)$, it was proved in [19] that, under some mean-reverting assumptions, such a process converges to its equilibrium at a rate of order $t^{-\alpha}$ where $\alpha \in (0,1)$ (depending on $H$). In [11], this result has been extended to the multiplicative case when $H>1/2$. In this paper, we obtain these types of results in the rough setting $H\in (1/3,1/2)$. Once again, we retrieve the rate orders of the additive setting. Our methods also extend the multiplicative results of [11] by deleting the gradient assumption on the noise coefficient $\sigma$. The main theorems include some existence and uniqueness results for the invariant distribution.
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https://hal.archives-ouvertes.fr/hal-01309650
Contributor : Aurélien Deya <>
Submitted on : Tuesday, October 4, 2016 - 9:11:35 AM
Last modification on : Tuesday, April 16, 2019 - 3:27:36 PM
Document(s) archivé(s) le : Thursday, January 5, 2017 - 12:54:42 PM

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• HAL Id : hal-01309650, version 2
• ARXIV : 1605.00880

### Citation

Aurélien Deya, Fabien Panloup, Samy Tindel. Rate of convergence to equilibrium of fractional driven stochastic differential equations with rough multiplicative noise. 2016. 〈hal-01309650v2〉

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