Optimal stopping of Hunt and Lévy processes

Abstract

Infinite horizon (perpetual) optimal stopping problems for Hunt processes on R are studied via the representation theory of excessive functions. In particular, we focus on problems with one-sided structure, that is, there exists a point x* such that the stopping region is of the form . The main result states that if it is possible to find a Radon measure such that the excessive function induced by this measure via the spectral representation has some very intuitive properties then the constructed excessive function coincides with the value function of the problem. Corresponding results for two-sided problems are also indicated. Specializing to Lévy processes, we obtain, by applying the Wiener–Hopf factorization, a general representation of the value function in terms of the maximum of the Lévy process. To illustrate the results, an explicit expression for the Green kernel of Brownian motion with exponential jumps is computed and some optimal stopping problems for Poisson process with positive exponential jumps and negative drift are solved.

Keywords:

• Optimal stopping problem
• Markov processes
• Hunt processes
• Lévy processes
• Green kernel
• Diffusions with jumps

AMS Classification::

• 60G40
• 60J25
• 60J30
• 60J60
• 60J75

Acknowledgements

Ernesto Mordecki thanks Åbo Akademi (Åbo, Finland), and the National Visitors Program (Finland) for the support and hospitality, that made possible the initiation of this work. The research of Paavo Salminen was supported by the Academy of Finland (grant no. 105849).

Notes

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Additional information

Notes on contributors

Ernesto Mordecki

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