Nonparametric estimation of linear multiplier for processes driven by a Hermite process

TL;DR

The paper tackles nonparametric estimation of the time-varying linear multiplier $\theta(t)$ in a stochastic differential equation driven by a Hermite process $Z_t^{q,H}$, in the small-noise limit $\varepsilon\to0$. It introduces a kernel-type estimator based on $\widehat{\theta}_t X_t=\frac{1}{\varphi_\varepsilon}\int G\left(\frac{\tau-t}{\varphi_\varepsilon}\right)dX_\tau$ and derives uniform consistency (Theorem 4.1), bias-variance_tradeoffs and optimal bandwidth for higher smoothness (Theorem 4.2), and the asymptotic distribution governed by the Hermite noise (Theorem 4.3). The analysis relies on properties of Hermite processes, including self-similarity, long-range dependence, and the covariance structure of Wiener integrals with respect to $Z^{q,H}$. An alternative estimator based on a censored process is also developed (Theorem 6.1), offering another route to nonparametric inference with explicit rate statements. Overall, the work extends nonparametric trend estimation to Hermite-driven SDEs, providing convergence guarantees and asymptotic characterizations under small-noise regimes relevant for long-range dependent, non-Gaussian settings.

Abstract

We study the problem of nonparametric estimation of the linear multiplier function $θ(t)$ for processes satisfying stochastic differential equations of the type $$dX_t=θ(t) X_tdt+ εdZ^{q,H}_t, X_0=x_0, 0\leq t \leq T$$ where $\{Z^{q,H}_t, t \geq 0\}$ is a Hermite process with known order $q$ and known self-similarity parameter $H \in (\frac{1}{2},1).$ We investigate the asymptotic behaviour of the estimator of the unknown function $θ(t)$ as $ε\rightarrow 0.$