Parameter estimation for nonergodic Ornstein-Uhlenbeck process driven by the weighted fractional Brownian motion

Abstract

Abstract In this paper, we consider the nonergodic Ornstein-Uhlenbeck process X 0 = 0 , d X t = θ X t d t + d B t a , b , t ≥ 0 , $$ X_{0}=0, \quad\quad dX_{t}=\theta X_{t} \,dt+dB_{t}^{a,b},\quad t\geq0, $$ driven by the weighted fractional Brownian motion B t a , b $B_{t}^{a,b}$ with parameter a and b. Our goal is to estimate the unknown parameter θ > 0 $\theta>0$ based on the discrete observations of the process. We construct two estimators θ ˆ n $\hat{\theta}_{n}$ and θ ˇ n $\check{\theta}_{n}$ of θ and show their strong consistency and the rate consistency.

Keywords

• parameter estimation
• Ornstein-Uhlenbeck process
• weighted fractional Brownian motion
• discrete observations