Estimation of several parameters in discretely-observed Stochastic Differential Equations with additive fractional noise

TL;DR

This paper develops a method for jointly estimating multiple parameters, including the Hurst parameter, in stochastic differential equations driven by fractional Brownian motion using discrete observations, with proven consistency and convergence.

Contribution

It introduces a novel estimator for several parameters in SDEs with fractional noise, based on stationary distribution assumptions, and validates it with theoretical and numerical results.

Findings

Estimator is consistent under stationarity assumptions

Convergence rate of the estimator is derived

Numerical experiments confirm theoretical properties

Abstract

We investigate the problem of joint statistical estimation of several parameters for a stochastic differential equation driven by an additive fractional Brownian motion. Based on discrete-time observations of the model, we construct an estimator of the Hurst parameter, the diffusion parameter and the drift, which lies in a parametrised family of coercive drift coefficients. Our procedure is based on the assumption that the stationary distribution of the SDE and of its increments permits to identify the parameters of the model. Under this assumption, we prove consistency results and derive a rate of convergence for the estimator. Finally, we show that the identifiability assumption is satisfied in the case of a family of fractional Ornstein-Uhlenbeck processes and illustrate our results with some numerical experiments.