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On the Continuous and Smooth Fit Principle for Optimal Stopping Problems in Spectrally Negative Lévy Models

Part of: Stochastic processes Markov processes

Published online by Cambridge University Press:  22 February 2016

Masahiko Egami  and
Kazutoshi Yamazaki
Masahiko Egami*
Affiliation:
Kyoto University
Kazutoshi Yamazaki*
Affiliation:
Kansai University
*
∗ Postal address: Graduate School of Economics, Kyoto University, Sakyo-Ku, Kyoto, 606-8501, Japan. Email address: egami@econ.kyoto-u.ac.jp
∗∗ Postal address: Department of Mathematics, Faculty of Engineering Science, Kansai University, Suita-shi, Osaka, 564-8680, Japan. Email address: kyamazak@kansai-u.ac.jp
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Abstract

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We consider a class of infinite time horizon optimal stopping problems for spectrally negative Lévy processes. Focusing on strategies of threshold type, we write explicit expressions for the corresponding expected payoff via the scale function, and further pursue optimal candidate threshold levels. We obtain and show the equivalence of the continuous/smooth fit condition and the first-order condition for maximization over threshold levels. As examples of its applications, we give a short proof of the McKean optimal stopping problem (perpetual American put option) and solve an extension to Egami and Yamazaki (2013).

Keywords

Optimal stoppingspectrally negative Lévy processscale functioncontinuous and smooth fit

MSC classification

Primary: 60G40: Stopping times; optimal stopping problems; gambling theory
Secondary: 60J75: Jump processes
Type
Research Article
Information
Advances in Applied Probability , Volume 46 , Issue 1 , March 2014 , pp. 139 - 167
Copyright
© Applied Probability Trust 

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