On drift parameter estimation for reflected fractional Ornstein–Uhlenbeck processes

Abstract

We consider a reflected Ornstein–Uhlenbeck process X driven by a fractional Brownian motion with Hurst parameter . Our goal is to estimate an unknown drift parameter on the basis of continuous observation of the state process. We establish Girsanov theorem for the process X, derive the standard maximum likelihood estimator of the drift parameter , and prove its strong consistency and asymptotic normality. As an improved estimator, we obtain the explicit formulas for the sequential standard maximum likelihood estimator and its mean squared error by assuming the process is observed until a certain information reaches a specified precision level. The estimator is shown to be unbiased, uniformly normally distributed, and efficient in the mean square error sense.

Keywords:

• Reflected fractional Ornstein–Uhlenbeck processes
• fractional Brownian motion
• fractional calculus
• parameter estimation
• maximum likelihood estimator
• sequential maximum likelihood estimator

AMS Subject Classifications:

• Primary: 60G22
• Secondary: 60H30
• 62M09
• 90B22

Acknowledgements

The authors would like to thank Professors David Nualart and Yaozhong Hu for their helpful comments and suggestions on an earlier draft of this paper. The authors also would like to thank two anonymous referees for their valuable comments which helped them to improve the paper.

Notes

No potential conflict of interest was reported by the authors.

Additional information

Funding

The research work of C. Lee is supported in part by Army Research Office [W911NF-14-1-0216].