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

TL;DR

This paper studies the rate at which ergodic stochastic differential equations driven by fractional Brownian motion with rough noise converge to equilibrium, extending previous results to the rougher Hurst parameter range and removing gradient assumptions.

Contribution

It extends convergence rate results to the rough Hurst parameter range (1/3, 1/2) and removes the gradient assumption on the noise coefficient, broadening the applicability of previous findings.

Findings

Convergence rates of order t^{-eta} for H in (1/3, 1/2)

Existence and uniqueness of invariant distributions

Extension of multiplicative noise results without gradient assumption

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 $H\textgreater{}1/2$. 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.