arXiv subjects
Publications and source records attributed to Maylis Varvenne.
We build and study a data-driven procedure for the estimation of the stationary density f of an additive fractional SDE. To this end, we also prove some new concentrations bounds for discrete observations of such dynamics in stationary regime.
In this paper we consider the drift estimation problem for a general differential equation driven by an additive multidimensional fractional Brownian motion, under ergodic assumptions on the drift coefficient. Our estimation procedure is based on the identification of the invariant measure, and we provide consistency results as well as some information about the convergence rate. We also give some examples of coefficients for which the identifiability assumption for the invariant measure is satisfied.
In this paper, we establish concentration inequalities both for functionals of the whole solution on an interval [0, T ] of an additive SDE driven by a fractional Brownian motion with Hurst parameter H $\in$ (0, 1) and for functionals of discrete-time observations of this process. Then, we apply this general result to specific functionals related to discrete and continuous-time occupation measures of the process.
The main objective of the paper is to study the long-time behavior of general discrete dynamics driven by an ergodic stationary Gaussian noise. In our main result, we prove existence and uniqueness of the invariant distribution and exhibit some upper-bounds on the rate of convergence to equilibrium in terms of the asymptotic behavior of the covariance function of the Gaussian noise (or equivalently to its moving average representation). Then, we apply our general results to fractional dynamics (including the Euler Scheme associated to fractional driven Stochastic Differential Equations). Whenthe Hurst parameter H belongs to (0, 1/2) we retrieve, with a slightly more explicit approach due to the discrete-time setting, the rate exhibited by Hairer in a continuous time setting. In this fractional setting, we also emphasize the significant dependence of the rate of convergence to equilibriumon the local behaviour of the covariance function of the Gaussian noise.