Optimal prediction of the ultimate maximum of Brownian motion

Abstract

At time 0 start to observe a Brownian path. Based upon the information, which is continuously updated through the observation of the path, a stopping time is determined such that the path is as close as possible to its unknown ultimate maximum over a finite time interval. The closeness is measured by a q-mean or by a probability distance. This can be formulated as an optimal stopping problem. The method of proof relies upon a representation of a conditional expectation of the gain process and the principle of smooth fit (at a single point).

Keywords:

• Brownian motion
• Ultimate maximum
• Optimal stopping
• Lévy's distributional theorem
• Smooth fit (at a single point)
• Viscosity solution
• Primary 60G40
• 60J65
• Secondary 60J60

Acknowledgements

The author is supported by a Steno grant from the Danish Natural Science Research Council, no. 21-02-0126