Nonparametric estimation of linear multiplier for processes driven by a bifractional Brownian motion

TL;DR

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

Contribution

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

Findings

Estimator's asymptotic behavior characterized as noise level approaches zero

Provides theoretical insights into the estimator's convergence properties

Enhances understanding of parameter estimation in complex stochastic processes

Abstract

We study the problem of nonparametric estimation of the linear multiplier function $\theta(t)$ for processes satisfying stochastic differential equations of the type $$dX_t=\theta(t)X_tdt+\epsilon dW_t^{H,K}, X_0=x_0,0\leq t \leq T$$ where $\{W_t^{H,K}, t \geq 0\}$ is a bifractional Brownian motion with known parameters $H\in (0,1), K\in (0,1]$ and $HK\in (\frac{1}{2},1).$ We investigate the asymptotic behaviour of the estimator of the unknown function $\theta(t)$ as $\epsilon \rightarrow 0.$