Annales de l'Institut Henri Poincaré, Probabilités et Statistiques

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

Joaquin Fontbona and Fabien Panloup

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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 $H>1/2$ and multiplicative noise component $\sigma$. When $\sigma$ is constant and for every $H\in(0,1)$, it was proved by Hairer 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 $H$). The aim of this paper is to extend such types of results to some multiplicative noise setting. More precisely, we show that we can recover such convergence rates when $H>1/2$ and the inverse of the diffusion coefficient $\sigma$ is a Jacobian matrix. The main novelty of this work is a type of extension of Foster–Lyapunov like techniques to this non-Markovian setting, which allows us to put in place an asymptotic coupling scheme without resorting to deterministic contracting properties.

Résumé

Cet article est consacré à la vitesse de convergence à l’équilibre pour des équations différentielles stochastiques multiplicatives dirigées par un mouvement brownien fractionnaire (fBm). Dans le cas additif, i.e. lorsque le coefficient « diffusif » $\sigma$ est constant et non dégénéré, cette question a été étudiée par Hairer qui, sous des hypothèses de contraction du coefficient de dérive en dehors d’un compact, a établi par des méthodes de couplage qu’un tel processus converge à l’équilibre à une vitesse dominée par $Ct^{-\alpha}$, où $\alpha\in(0,1)$ dépend de l’indice de Hurst $H$ du fBm. L’objectif de notre travail est d’étendre ce type de résultat au cadre multiplicatif. Plus précisément, nous montrons que si $H>1/2$ et si $\sigma^{-1}$ est une matrice jacobienne, alors le résultat précédent reste vrai avec des bornes identiques sur la vitesse de convergence. La principale nouveauté de ce travail réside dans le développement de techniques de type Foster–Lyapounov dans ce cadre non markovien, nous permettant de mettre en place un schéma de couplage similaire à [9] sans faire appel à des propriétés de contraction déterministes.

Article information

Source
Ann. Inst. H. Poincaré Probab. Statist., Volume 53, Number 2 (2017), 503-538.

Dates
Received: 12 June 2014
Revised: 1 August 2015
Accepted: 21 October 2015
First available in Project Euclid: 11 April 2017

Permanent link to this document
https://projecteuclid.org/euclid.aihp/1491897735

Digital Object Identifier
doi:10.1214/15-AIHP724

Mathematical Reviews number (MathSciNet)
MR3634264

Zentralblatt MATH identifier
1367.60067

Subjects
Primary: 60G22: Fractional processes, including fractional Brownian motion 37A25: Ergodicity, mixing, rates of mixing

Keywords
Stochastic differential equations Fractional Brownian motion Multiplicative noise Ergodicity Rate of convergence to equilibrium Lyapunov function Total variation distance

Citation

Fontbona, Joaquin; Panloup, Fabien. Rate of convergence to equilibrium of fractional driven stochastic differential equations with some multiplicative noise. Ann. Inst. H. Poincaré Probab. Statist. 53 (2017), no. 2, 503--538. doi:10.1214/15-AIHP724. https://projecteuclid.org/euclid.aihp/1491897735


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