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 θ ∈ R in the linear modelwhere B H 1 and B H 2 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.Keywords Fractional Brownian motion, maximum likelihood estimator, integral equation with weakly singular kernel, compact operator, asymptotic consistency 2010 MSC 60G22, 62F10, 62F12, 62G12

Parameter Estimation for Gaussian Processes with Application to the Model with Two Independent Fractional Brownian Motions

“…⊓ ⊔ are also bounded and defined on the entire space L 2 [0, T ]. By (34) and the semigroup property (Theorem 9),…”

“…whence the inequality (37) follows. The second statement is obtained by the representation (34). ⊓ ⊔ are also bounded and defined on the entire space L 2 [0, T ].…”

“…Therefore Remark 2. Another approach to the drift parameter estimation in the model with two fractional Brownian motions was proposed in [28] and developed in [34]. It is based on the solving of the following Fredholm integral equation of the second kind…”

“…This model was first studied in [28], where a strongly consistent estimator for the unknown drift parameter θ was constructed for 1/2 < H 1 < H 2 < 1 and H 2 − H 1 > 1/4 by continuous-time observations of X. Later, in [34], the strong consistency of this estimator was proved for arbitrary 1/2 < H 1 < H 2 < 1. The details on this approach are given in Remark 2 below.…”

Maximum Likelihood Drift Estimation for Gaussian Process with Stationary Increments

“…However, on the one hand, our approach is different from their one, it cannot be deduced from their general formulas and on the other hand, gives rather elegant representations. The construction of the maximum likelihood estimator in the case when B is the sum of two fractional Brownian motions was studied in Mishura (2016) and Mishura and Voronov (2015). A similar non-Gaussian model driven by the Rosenblatt process was considered in Bertin et al (2011).…”