Maximum likelihood drift estimation for the mixing of two fractional Brownian motions

We construct the maximum likelihood estimator (MLE) of the unknown drift parameter $\theta \in \mathbb{R}$ in the linear model $X_t=\theta t+\sigma B^{H_1}(t)+B^{H_2}(t),;t\in[0,T],$where$B^{H_1}$and$B^{H_2}$ are two independent fractional Brownian motions with Hurst indices $\frac{1}{2}<H_1<H_2<1.$ The formula for MLE is based on the solution of the integral equation with weak polar kernel.