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).
Optimal stopping Optimal stopping boundary Principle of smooth fit Principle of continuous fit Lévy processes Scale functions
Mathematics Subject Classification
60G40 60G51 60J75
Alili, L., Kyprianou, A.E.: Some remarks of first passage of Lévy processes, the American put and pasting principles. Ann. Appl. Probab. 15, 2062–2080 (2005)
Avram, F., Kyprianou, A.E., Pistorius, M.R.: Exit problems for spectrally negative Lévy processes and applications to (Canadized) Russian options. Ann. Appl. Probab. 14, 215–238 (2004)
Conze, A., Viswanathan, R.: Path dependent options: the case of lookback options. J. Finance 46, 1893–1907 (1991)
Cox, A.M.G., Hobson, D., Obłój, J.: Pathwise inequalities for local time: applications to Skorohod embeddings and optimal stopping. Ann. Appl. Probab. 18, 1870–1896 (2008)