# A Capped Optimal Stopping Problem for the Maximum Process

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## 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## References

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