On drift parameter estimation in models with fractional Brownian motion

Abstract

We consider a stochastic differential equation involving standard and fractional Brownian motion with unknown drift parameter to be estimated. We investigate the standard maximum likelihood estimate of the drift parameter, two non-standard estimates and three estimates for the sequential estimation. Model strong consistency and some other properties are proved. The linear model and Ornstein–Uhlenbeck model are studied in detail. As an auxiliary result, an asymptotic behaviour of the fractional derivative of the fractional Brownian motion is established.

Keywords:

• fractional Brownian motion
• Brownian motion
• parameter estimation
• stochastic differential equation
• sequential estimation

AMS Subject Classification:

• 60G22
• 60J65
• 60H10
• 62F05

Acknowledgements

This paper was partially supported by NSERC grant 261855. We are thankful to Ivan Smirnov and Georgii Shevchenko for the assistance in the preparation of the manuscript. We are thankful to the anonymous referees for their helpful suggestions.