Construction of maximum likelihood estimator in the mixed fractional--fractional Brownian motion model with double long-range dependence

Abstract: We construct an estimator of the unknown drift parameter $\theta\in {\mathbb{R}}$ in the linear model [X_t=\theta t+\sigma_1B^{H_1}(t)+\sigma_2B^{H_2}(t),\;t\in[0,T],] where $B^{H_1}$ and $B^{H_2}$ are two independent fractional Brownian motions with Hurst indices $H_1$ and $H_2$ satisfying the condition $\frac{1}{2}\leq H_1<H_2<1.$ Actually, we reduce the problem to the solution of the integral Fredholm equation of the 2nd kind with a specific weakly singular kernel depending on two power exponents. It is proved that the kernel can be presented as the product of a bounded continuous multiplier and weak singular one, and this representation allows us to prove the compactness of the corresponding integral operator. This, in turn, allows us to establish an existence--uniqueness result for the sequence of the equations on the increasing intervals, to construct accordingly a sequence of statistical estimators, and to establish asymptotic consistency.