Parameter estimation for fractional mixed fractional Brownian motion based on discrete observations

Abstract

The object of investigation is the mixed fractional Brownian motion of the form ${X_{t}}=\kappa {B_{t}^{{H_{1}}}}+\sigma {B_{t}^{{H_{2}}}}$, driven by two independent fractional Brownian motions ${B_{1}^{H}}$ and ${B_{2}^{H}}$ with Hurst parameters ${H_{1}}\lt {H_{2}}$. Strongly consistent estimators of unknown model parameters ${({H_{1}},{H_{2}},{\kappa ^{2}},{\sigma ^{2}})^{\top }}$ are constructed based on the equidistant observations of a trajectory. Joint asymptotic normality of these estimators is proved for $0\lt {H_{1}}\lt {H_{2}}\lt \frac{3}{4}$.

Keywords

• fractional Brownian motion
• mixed model
• strong consistency
• ergodic theorem
• Asymptotic normality