An optimal stopping problem for spectrally negative Markov additive processes

STOCHASTIC PROCESSES AND THEIR APPLICATIONS

卷:150

Article

Caglar, M.¹  Kyprianou, A.²  Vardar-Acar, C.³ 

[1] Koc Univ, Coll Sci, Dept Math, Rumeli Feneri Yolu, TR-34450 Istanbul, Turkey

[2] Univ Bath, Dept Math Sci, Claverton Down BA2 7AY, England

[3] Middle East Tech Univ, Dept Stat, Universiteler Mah,Dumlupinar Blv 1, TR-06800 Cankaya, Turkey

关键词: Optimal stopping;    Scale matrices;    Excursion theory;    Markov additive processes;   

DOI  :  10.1016/j.spa.2021.06.010

来源: Elsevier

【 摘 要 】

Previous authors have considered optimal stopping problems driven by the running maximum of a spectrally negative Levy process as well as of a one-dimensional diffusion; see e.g. Kyprianou and Ott (2014); Ott (2014); Ott (2013); Alvarez and Matomaki (2014); Guo and Shepp (2001); Pedersen (2000); Gapeev (2007). Many of the aforementioned results are either implicitly or explicitly dependent on Peskir's maximality principle, cf. (Peskir, 1998). In this article, we are interested in understanding how some of the main ideas from these previous works can be brought into the setting of problems driven by the maximum of a class of Markov additive processes (more precisely Markov modulated Levy processes). Similarly to Ott (2013); Kyprianou and Ott (2014); Ott (2014), the optimal stopping boundary is characterised by a system of ordinary first-order differential equations, one for each state of the modulating component of the Markov additive process. Moreover, whereas scale functions played an important role in the previously mentioned work, we work instead with scale matrices for Markov additive processes here; as introduced by Kyprianou and Palmowski (2008); Ivanovs and Palmowski (2012). We exemplify our calculations in the setting of the Shepp-Shiryaev optimal stopping problem (Shepp and Shiryaev, 1993; Shepp and Shiryaev, 1995), as well as a family of capped maximum optimal stopping problems. (C) 2021 Elsevier B.V. All rights reserved.

【 授权许可】

Free