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

Volume 2, Issue 2 (2015), pp. 147–164

Yuliya Mishura

^{ }Ivan Voronov^{ }
Pub. online: 20 July 2015
Type: Research Article
Open Access

Received

17 June 2015

17 June 2015

Revised

7 July 2015

7 July 2015

Accepted

7 July 2015

7 July 2015

Published

20 July 2015

20 July 2015

#### Abstract

We construct an estimator of the unknown drift parameter $\theta \in \mathbb{R}$ in the linear model
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 $\frac{1}{2}\le H_{1}<H_{2}<1$. 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.

#### References

Jost, C.: Transformation formulas for fractional Brownian motion. Stochastic Processes and their Applications 116(10), 1341–1357 (2006) MR2260738. doi:10.1016/j.spa.2006.02.006

Mishura, Y.: Maximum likelihood drift estimation for the mixing of two fractional Brownian motions. arXiv preprint arXiv:1506.04731v1 (2015)