On predicting the ultimate maximum for exponential Lévy processes

Abstract

;We consider a problem of predicting of the ultimate maximum of the process over a finite interval of time. Mathematically, this problem relates to a particular optimal stopping problem. In this paper we discuss exponential Lévy processes. As the Lévy processes, we discuss $\alpha$-stable Lévy processes, $0<\alpha\leq 2$, and generalized hyperbolic Lévy processes. The method of solution uses the representations of these processes as time-changed Brownian motions with drift. Our results generalize results of papers by Toit and Peskir and by Shiryaev and Xu, and Zhou.