Parameter Estimation for Brownian Motion with Trend Mixed with Two Fractional Brownian Motions

Abstract

This work studies parameter estimation for a stochastic process that combines a Brownian motion with trend and two fractional Brownian motions with distinct Hurst indices. The process is given by , where is a standard Brownian motion and are fractional Brownian motions with Hurst indices such that , with ,  and being mutually independent. An ergodic theorem is applied to derive estimators for all unknown parameters and examine the effect of different Borel function selections on the estimator formulations. Numerical simulations are performed to confirm the asymptotic properties of the estimators and to assess their accuracy under various configurations. The estimators are then applied to cryptocurrency data to demonstrate their practical use.