Acta Applicandae Mathematicae

, Volume 129, Issue 1, pp 147–174 | Cite as

A Capped Optimal Stopping Problem for the Maximum Process

Article

Abstract

This paper concerns an optimal stopping problem driven by the running maximum of a spectrally negative Lévy process X. More precisely, we are interested in capped versions of the American lookback optimal stopping problem (Gapeev in J. Appl. Probab. 44:713–731, 2007; Guo and Shepp in J. Appl. Probab. 38:647–658, 2001; Pedersen in J. Appl. Probab. 37:972–983, 2000), which has its origins in mathematical finance, and provide semi-explicit solutions in terms of scale functions. The optimal stopping boundary is characterised by an ordinary first-order differential equation involving scale functions and, in particular, changes according to the path variation of X. Furthermore, we will link these capped problems to Peskir’s maximality principle (Peskir in Ann. Probab. 26:1614–1640, 1998).

Keywords

Optimal stopping Optimal stopping boundary Principle of smooth fit Principle of continuous fit Lévy processes Scale functions 

Mathematics Subject Classification

60G40 60G51 60J75 

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Copyright information

© Springer Science+Business Media Dordrecht 2013

Authors and Affiliations

  1. 1.Mathematical SciencesUniversity of BathBathUK

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