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

Abstract

We study the problem of nonparametric estimation of the linear multiplier function $\hat{I}¸(t)$ for processes satisfying stochastic differential equations of the type $$d{X}_t=\hat{I}¸(t)X_{t}dt+\hat{I}\mu d{W}_t^{H,K},X_0=x_0,0\le t\le T$$ where $\{W_t^{H,K},t\ge 0\}$ 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 $\hat{I}¸(t)$ as $\hat{I}\mu \to 0.$