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Article Dans Une Revue Annals of Probability Année : 2019

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

Aurélien Deya
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Samy Tindel
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Résumé

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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Dates et versions

hal-01309650 , version 1 (29-04-2016)
hal-01309650 , version 2 (04-10-2016)
hal-01309650 , version 3 (27-02-2019)

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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. Annals of Probability, 2019, 47 (1), pp.464-518. ⟨10.1214/18-aop1265⟩. ⟨hal-01309650v3⟩
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