A capped optimal stopping problem for the maximum process

Andreas Kyprianou, Curdin Ott

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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).
Original languageEnglish
Pages (from-to)147-174
Number of pages28
JournalActa Applicandae Mathematicae
Issue number1
Early online date8 Jun 2013
Publication statusPublished - Feb 2014


  • Levy processes
  • optimal stopping
  • optimal stopping boundary
  • principle of continuous fit
  • principle of smooth fit
  • scale functions


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