Drift estimation for rough processes under small noise asymptotic : trajectory fitting method

Abstract

We consider a process $X^\ve$ that solves a stochastic Volterra equation with an unknown parameter $\hat{I}{¸}^{\star }$ in the drift function. The Volterra kernel is singular, and includes as an example, $K\_0(u)=c u^{α-1/2} \id{u>0}$ with $\hat{I}\pm \in (0,1/2)$. It is assumed that the diffusion coefficient is proportional to $\ve \to 0$. From an observation of the path $(X^\ve\_s)\_{s\in[0,T]}$, we construct a Trajectory Fitting Estimator, which is shown to be consistent and asymptotically normal. We also specify identifiability conditions insuring the $L^p$ convergence of the estimator.