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

We consider a process X^$\epsilon$ solution of a stochastic Volterra equation with an unknown parameter $\theta$ in the drift function. The Volterra kernel is singular and given by K(u) = cu $\alpha$-1 __u>0 with $\alpha$ $\in$ (1/2, 1) and it is assumed that the diffusion coefficient is proportional to $\epsilon$ $\rightarrow$ 0 Based on the observation of a discrete sampling with mesh h $\rightarrow$ 0 of the Volterra process, we build a Quasi Maximum Likelihood Estimator. The main step is to assess the error arising in the reconstruction of the path of a semi-martingale from the inversion of the Volterra kernel. We show that this error decreases as h^{1/2} whatever is the value of $\alpha$. Then, we can introduce an explicit contrast function, which yields an efficient estimator when $\epsilon$ $\rightarrow$ 0.