A Capped Optimal Stopping Problem for the Maximum Process
This paper concerns an optimal stopping problem driven by the running maximum of a spectrally negative Lvy process X. More precisely, we are interested in capped versions of the American lookback optimal stopping problem (Gapeev in J.Appl. Probab. 44:713731, 2007; Guo and Shepp in J.Appl. Probab. 38:647658, 2001; Pedersen in J.Appl. Probab. 37:972983, 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 Peskirs maximality principle (Peskir in Ann. Probab. 26:16141640, 1998).
optimal stopping lvy processes scale functions optimal stopping boundary principle of smooth fit principle of continuous fit 60g40 60g51 60j75