Inference for SDEs driven by Hermite processes

Abstract: In the paper, we address parametric and non-parametric estimation for nonlinear stochastic differential equations with additive Hermite noise with possibly nonlinear scaling. We assume that a single trajectory of the solution is observed discretely and we propose estimators of the Hurst parameter and the Hermite order of the driving process as well as of the average noise intensity and noise intensity function. The estimators are based on the weighted quadratic variation whose properties are used, in particular, to prove weak consistency of the proposed estimators under in-fill asymptotics.