Maximum Likelihood Drift Estimation for the Mixing of Two Fractional Brownian Motions

Abstract

We construct the maximum likelihood estimator (MLE) of the unknown drift parameter _ θ ∈ R in the linear model Xt θt + σ1 BH1(t) + σ2 BH2(t), t ∈ [0,T], where BH1 and BH2 are two independent fractional Brownian motions with Hurst indices 1/2 ≤ H1 < H2 < 1: The formula for MLE is based on the solution of the integral equation with weak polar kernel.