Inference for fractional Ornstein-Uhlenbeck type processes with periodic mean in the non-ergodic case

Abstract

In the paper we consider the problem of estimating parameters entering the drift of a fractional Ornstein-Uhlenbeck type process in the non-ergodic case, when the underlying stochastic integral is of Young type. We consider the sampling scheme that the process is observed continuously on $[0,T]$ and $T\to \infty .$ For known Hurst parameter $H\in (0.5,1),$ i.e. the long range dependent case, we construct a least-squares type estimator and establish strong consistency. Furthermore, we prove a second order limit theorem which provides asymptotic normality for the parameters of the periodic function with a rate depending on H and a non-central Cauchy limit result for the mean reverting parameter with exponential rate. For the special case that the periodicity parameter is the weight of a periodic function, which integrates to zero over the period, we can even improve the rate to $\sqrt{T}.$

Keywords:

• Fractional Ornstein Uhlenbeck process
• long range dependence
• periodic mean function
• least squares estimator
• non-ergodicity

2010 MATHEMATICS SUBJECT CLASSIFICATION:

• Primary 62M09
• Secondary 60G22
• 60H10

Acknowledgements

The financial support of the DFG-SFB 823 is gratefully acknowledged.

Additional information

Funding

German Research Foundation.