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

TL;DR

This paper develops a nonparametric estimator for the linear multiplier function in stochastic differential equations driven by Hermite processes, analyzing its asymptotic behavior as noise diminishes.

Contribution

It introduces a novel nonparametric estimation method for the multiplier function in Hermite-driven SDEs and studies its asymptotic properties.

Findings

Estimator's asymptotic behavior characterized as noise approaches zero

Provides theoretical insights into the estimator's convergence

Enhances understanding of parameter estimation in Hermite process models

Abstract

We study the problem of nonparametric estimation of the linear multiplier function $\theta(t)$ for processes satisfying stochastic differential equations of the type $$dX_t=\theta(t) X_tdt+ \epsilon dZ^{q,H}_t, X_0=x_0, 0\leq t \leq T$$ 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 $\theta(t)$ as $\epsilon \rightarrow 0.$