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