Optimal stopping problems for the maximum process with upper and lower caps
Abstract
This paper concerns optimal stopping problems driven by the running maximum of a spectrally negative Lévy process $X$. More precisely, we are interested in modifications of the SheppShiryaev optimal stopping problem [Avram, Kyprianou and Pistorius Ann. Appl. Probab. 14 (2004) 215238; Shepp and Shiryaev Ann. Appl. Probab. 3 (1993) 631640; Shepp and Shiryaev Theory Probab. Appl. 39 (1993) 103119]. First, we consider a capped version of the SheppShiryaev optimal stopping problem and provide the solution explicitly in terms of scale functions. In particular, the optimal stopping boundary is characterised by an ordinary differential equation involving scale functions and changes according to the path variation of $X$. Secondly, in the spirit of [Shepp, Shiryaev and Sulem Advances in Finance and Stochastics (2002) 271284 Springer], we consider a modification of the capped version of the SheppShiryaev optimal stopping problem in the sense that the decision to stop has to be made before the process $X$ falls below a given level.
 Publication:

arXiv eprints
 Pub Date:
 July 2011
 DOI:
 10.48550/arXiv.1107.0233
 arXiv:
 arXiv:1107.0233
 Bibcode:
 2011arXiv1107.0233O
 Keywords:

 Mathematics  Probability
 EPrint:
 Published in at http://dx.doi.org/10.1214/12AAP903 the Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute of Mathematical Statistics (http://www.imstat.org)