Abstract
We study optimal stopping of strong Markov processes under random implementation delay. By random implementation delay we mean the following: the payoff is not realised immediately when the process is stopped but rather after a random waiting period. The distribution of the random waiting period is assumed to be phase-type. We prove first a general result on the solvability of the problem. Then we study the case of Coxian distribution both in general and with scalar diffusion dynamics in more detail. The study is concluded with two explicit examples.
Highlights
Optimal stopping problems are widely used in economic and financial applications
Phase-type distributions have been broadly applied in different fields such as survival analysis (Aalen 1995), healthcare systems modelling (Fackrell 2009), insurance applications (Bladt 2005), queuing theory (Breure and Baum 2005), and population genetics (Hobolth et al 2018)
These distributions are a class of matrix exponential distributions that have a Markovian realisation: they can be identified as the absorption times of certain continuous time Markov chains
Summary
Optimal stopping problems are widely used in economic and financial applications. One of the most notable application is the real options approach to investment planning, for a textbook treatment of the topic, see, e.g., Dixit and Pindyck (1994). The effect of time to build for a levered firm is studied in Sarkar and Zhang (2015) It is assumed in this paper, in the style of Margsiri et al (2008), that a fixed proportion of the investment cost is payed at time when the investment is engaged and the rest is paid once the implementation period is elapsed. Phase-type distributions have been broadly applied in different fields such as survival analysis (Aalen 1995), healthcare systems modelling (Fackrell 2009), insurance applications (Bladt 2005), queuing theory (Breure and Baum 2005), and population genetics (Hobolth et al 2018) These distributions are a class of matrix exponential distributions that have a Markovian realisation: they can be identified as the absorption times of certain continuous time Markov chains.
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