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

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 near zero, exhibiting a behavior comparable to 𝐾 0 ⁡ ( 𝑢 ) = 𝑐 ⁢ 𝑢 𝛼 − 1 ⁢ 𝟙 𝑢 > 0 with α ∈ (1/2, 1). It is assumed that the diffusion coefficient is proportional to ε → 0. Based on discrete observations, with a mesh size h → 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 α . Then, we can introduce an explicit contrast function, which yields an efficient estimator when ε → 0.