Diffusion approximations for periodically arriving expert opinions in a financial market with Gaussian drift

Abstract

In this paper we study a financial market in which stock returns depend on an unobservable Gaussian drift process. Investors obtain information on that drift from return observations and discrete-time expert opinions as an external source of information. Estimates of the hidden drift process are based on filtering techniques. Our focus is the case of high-frequency experts periodically providing their views on the drift with variances growing linearly with the arrival frequency. The latter condition guarantees that the delivered information per time is limited. The asymptotic behavior of the filter as the arrival frequency tends to infinity is described by limit theorems. These state that the information obtained from observing the discrete-time expert opinions is asymptotically the same as that from observing a certain diffusion process. We apply these diffusion approximations of the filter for deriving simplified approximate solutions of utility maximization problems with logarithmic and power utility.

Keywords:

• Diffusion approximations
• Kalman–Bucy filter
• Ornstein–Uhlenbeck process
• expert opinions
• portfolio optimization
• partial information

2010 MSC:

• Primary 91G10
• Secondary 93E11
• 93E20
• 60F25

Acknowledgments

The authors thank Abdelali Gabih (Chouaib Doukkali University El Jadida) and Hakam Kondakji (Helmut Schmidt University Hamburg) for valuable discussions that improved this paper.