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

الأصل · EN

We construct an estimator of the unknown drift parameter θ∈ R in the linear model Xₜ=θt+σ₁Bʰ¹(t)+σ₂Bʰ²(t),t∈[0,T], where Bʰ¹ and Bʰ² are two independent fractional Brownian motions with Hurst indices H₁ and H₂ satisfying the condition 1/2≤ H₁<H₂<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.