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Resolvent-Techniques for Multiple Exercise Problems

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Abstract

We study optimal multiple stopping of strong Markov processes with random refraction periods. The refraction periods are assumed to be exponentially distributed with a common rate and independent of the underlying dynamics. Our main tool is using the resolvent operator. In the first part, we reduce infinite stopping problems to ordinary ones in a general strong Markov setting. This leads to explicit solutions for wide classes of such problems. Starting from this result, we analyze problems with finitely many exercise rights and explain solution methods for some classes of problems with underlying Lévy and diffusion processes, where the optimal characteristics of the problems can be identified more explicitly. We illustrate the main results with explicit examples.

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Notes

  1. under appropriate smoothness assumptions on \(g\), this means that \(\tilde{f}=(r-\mathcal {A})g\)

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Acknowledgments

We thank the anonymous referee for carefully reading earlier versions of this manuscript and for thoughtful and valuable comments that helped to improve the presentation of this article. Jukka Lempa acknowledges financial support from the project “Energy markets: modelling, optimization and simulation (EMMOS)”, funded by the Norwegian Research Council under Grant 205328.

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Authors and Affiliations

  1. Mathematical Institute, Christian–Albrechts-University in Kiel, Ludewig-Meyn-Str. 4, D-24098 , Kiel, Germany

    Sören Christensen

  2. Department of Mathematics, SPST, University of Hamburg, Bundesstrasse 55, 20146 , Hamburg, Germany

    Sören Christensen

  3. School of business, Faculty of Social Sciences, Oslo and Akershus University College, St. Olavs plass, P.O. Box 4 , 0130 , Oslo, Norway

    Jukka Lempa

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  1. Sören Christensen
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Correspondence to Jukka Lempa.

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Christensen, S., Lempa, J. Resolvent-Techniques for Multiple Exercise Problems. Appl Math Optim 71, 95–123 (2015). https://doi.org/10.1007/s00245-014-9254-4

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