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January 2019 Rate of convergence to equilibrium of fractional driven stochastic differential equations with rough multiplicative noise
Aurélien Deya, Fabien Panloup, Samy Tindel
Ann. Probab. 47(1): 464-518 (January 2019). DOI: 10.1214/18-AOP1265

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 [Ann. Probab. 33 (2005) 703–758] 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 [Ann. Inst. Henri Poincaré Probab. Stat. 53 (2017) 503–538], 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 [Ann. Inst. Henri Poincaré Probab. Stat. 53 (2017) 503–538] 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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Aurélien Deya. Fabien Panloup. Samy Tindel. "Rate of convergence to equilibrium of fractional driven stochastic differential equations with rough multiplicative noise." Ann. Probab. 47 (1) 464 - 518, January 2019. https://doi.org/10.1214/18-AOP1265

Information

Received: 1 October 2016; Revised: 1 November 2017; Published: January 2019
First available in Project Euclid: 13 December 2018

zbMATH: 07036342
MathSciNet: MR3909974
Digital Object Identifier: 10.1214/18-AOP1265

Subjects:
Primary: 37A25, 60G22

Rights: Copyright © 2019 Institute of Mathematical Statistics

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Vol.47 • No. 1 • January 2019
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