A capped optimal stopping problem for the maximum process

TL;DR

This paper studies a capped optimal stopping problem based on the maximum of a spectrally negative Levy process, providing semi-explicit solutions and characterizing the stopping boundary through differential equations.

Contribution

It introduces capped versions of the American lookback problem, linking solutions to scale functions and Peskir's maximality principle, with explicit boundary characterizations.

Findings

Semi-explicit solutions in terms of scale functions

Characterization of stopping boundary via differential equations

Connection to Peskir's maximality principle

Abstract

This paper concerns an optimal stopping problem driven by the running maximum of a spectrally negative Levy process X. More precisely, we are interested in capped versions of the American lookback optimal stopping problem, 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.