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

We consider a process $X^{\text{ve}}$ that solves a stochastic Volterra equation with an unknown parameter ${\theta }^{\star }$ in the drift function. The Volterra kernel is singular, and includes as an example, $K\mathrm{\_}0(u)=c{u}^{\alpha -1\mathrm{/}2}\text{id}u>0$ with $\alpha \in (0,1\mathrm{/}2)$. It is assumed that the diffusion coefficient is proportional to $\text{ve}\to 0$. From an observation of the path $(X^{\text{ve}}\mathrm{\_}s)\mathrm{\_}s\in [0,T]$, 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.