OPTIMAL STOPPING PROBLEMS FOR THE MAXIMUM PROCESS WITH UPPER AND LOWER CAPS

• 成果类型: Article
• 署名作者: Ott, Curdin
• 署名单位: University of Bath
• 刊物名称: ANNALS OF APPLIED PROBABILITY
• ISSN/ISSBN: 1050-5164
• DOI: 10.1214/12-AAP903
• 发表日期: 2013
• 页码: 2327-2356
• 关键词: Levy processes; driven; EXIT

摘要

This paper concerns optimal stopping problems driven by the running maximum of a spectrally negative Levy process X. More precisely, we are interested in modifications of the Shepp-Shiryaev optimal stopping problem [Avram, Kyprianou and Pistorius Ann. Appl. Probab. 14 (2004) 215-238; Shepp and Shiryaev Ann. Appl. Probab. 3 (1993) 631-640; Shepp and Shiryaev Theory Probab. Appl. 39 (1993) 103-119]. First, we consider a capped version of the Shepp-Shiryaev 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) 271-284 Springer], we consider a modification of the capped version of the Shepp-Shiryaev optimal stopping problem in the sense that the decision to stop has to be made before the process X falls below a given level.