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

TL;DR

This work studies drift parameter estimation for a rough stochastic Volterra equation with a singular kernel in the small-noise regime. It proposes a Trajectory Fitting Estimator based on the contrast $Q_\varepsilon(\theta)$ that compares the noisy path to a noise-free trajectory, proving consistency and asymptotic normality under explicit identifiability and regularity conditions. The analysis decomposes the contrast into drift and diffusion contributions, establishes uniform $L^p$ convergence of the contrast and its gradient, and derives a first-order expansion leading to a rate-$1/\varepsilon$ asymptotic distribution involving a linear Volterra limit $Z^0$ and a stochastic integral term. The results rely on resolvent relations for the kernel, Lipschitz continuity, and smooth dependence on the parameter, with an appendix detailing the differentiability of the deterministic flow. Practically, the findings enable reliable drift parameter estimation for rough volatility-type models in the small-noise regime, with rigorous finite-sample control via $L^p$ bounds and explicit asymptotic characterizations.

Abstract

We consider a process $X^\varepsilon$ solution of a stochastic Volterra equation with an unknown parameter $θ^\star$ in the drift function. The Volterra kernel is singular and given by $K(u)=c u^{α-1/2} \mathbb{1}_{u>0}$ with $α\in (0,1/2)$. It is assumed that the diffusion coefficient is proportional to $\varepsilon \to 0$. From an observation of the path $(X^\varepsilon_s)_{s\in[0,T]}$, 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.

Theorems & Definitions (21)

• Lemma 2.1
• proof
• Proposition 3.1
• Lemma 3.2
• proof
• Lemma 3.3
• proof
• Lemma 3.4
• proof
• proof