Nonparametric Estimation of Linear Multiplier for Processes Driven by Mixed fractional Brownian Motion

TL;DR

This paper investigates a nonparametric method to estimate a linear multiplier function in stochastic differential equations driven by mixed fractional Brownian motion, analyzing the estimator's asymptotic behavior as noise diminishes.

Contribution

It introduces a new nonparametric estimator for the linear multiplier in processes driven by mixed fractional Brownian motion and studies its asymptotic properties.

Findings

Estimator's asymptotic behavior characterized as noise level approaches zero

Provides theoretical foundation for nonparametric estimation in mixed fractional Brownian motion models

Enhances understanding of stochastic differential equations with fractional noise

Abstract

We study the problem of nonparametric estimation of linear multiplier function $\theta t)$ for processes satisfying stochastic differential equations of the type $dX_t=\theta(t)X_tdt+\epsilond\bar W_t^H, X_0=x_0, 0\leq t \leq T$ where $\{\bar W_T^H, t \geq 0\}$ is a mixed fractional Brownian motion with known Hurst index $H$ and study the asymptotic behaviour of the estimator as $\epsilon \rightarrow 0.$