On predicting the ultimate maximum for exponential Lévy processes

Katsunori Ano (Shibaura Institute of Technology, Tokyo)
Roman Ivanov (Trapeznikov Institute of Control Sciences of RAS, Moscow)

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.


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Pages: 1-9

Publication Date: October 7, 2012

DOI: 10.1214/ECP.v17-1805

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