Drift estimation for rough processes under small noise asymptotic : QMLE approach
Arnaud Gloter, Nakahiro Yoshida
TL;DR
The paper tackles drift estimation for rough Volterra-type SDEs with singular kernels under small-noise asymptotics using a Quasi-Maximum Likelihood framework. It introduces a discrete inversion scheme to reconstruct the latent semimartingale from finite-sample observations and builds an explicit contrast that yields an asymptotically Gaussian estimator with rate $\varepsilon^{-1}$, provided the sampling step $h$ is sufficiently small. A key result is that the reconstruction error scales as $h^{1/2}$ (up to a drift-term error of order $h^{\alpha}$), enabling reliable inference from discrete data. Theoretical findings are complemented by numerical simulations illustrating finite-sample performance and the influence of the tuning parameter $k$, with discussion on the roles of $\alpha$, $T$, and the Fisher-information structure in the resulting inference.
Abstract
We consider a process X^$ε$ solution of a stochastic Volterra equation with an unknown parameter $θ$ in the drift function. The Volterra kernel is singular and given by K(u) = cu $α$-1 __u>0 with $α$ $\in$ (1/2, 1) and it is assumed that the diffusion coefficient is proportional to $ε$ $\rightarrow$ 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 $α$. Then, we can introduce an explicit contrast function, which yields an efficient estimator when $ε$ $\rightarrow$ 0.