Drift estimation for rough processes under small noise asymptotic : QMLE approach

Abstract

We consider a process $X^\ve$ solution of a stochastic Volterra equation with an unknown parameter $\hat{I}{¸}^{\star }$ in the drift function. The Volterra kernel is singular near zero, exhibiting a behavior comparable to $K\_0(u)=cu^{α-1} \id{u>0}$ with $\hat{I}\pm \in (1/2,1)$.It is assumed that the diffusion coefficient is proportional to $\ve \to 0$. Based on discrete observations, with a mesh size $h\to 0$, of the Volterra process, we construct a Quasi Maximum Likelihood Estimator. The main step is to assess the error arising in the reconstruction of the path of a semimartingale from the inversion of the Volterra kernel. We show that this error decreases as $h^{1/2}$ regardless of the value of $\hat{I}\pm$. Then, we can introduce an explicit contrast function, which yields an efficient estimator when $\ve \to 0$.