An optimal stopping problem for spectrally negative Markov additive processes

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