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2014 | OriginalPaper | Buchkapitel

11. Applications to Optimal Stopping Problems

verfasst von : Andreas E. Kyprianou

Erschienen in: Fluctuations of Lévy Processes with Applications

Verlag: Springer Berlin Heidelberg

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Abstract

The aim of this chapter is to show how some of the established fluctuation identities for (reflected) Lévy processes can be used to solve quite specific, but nonetheless exemplary, optimal stopping problems. To some extent, this will be done in an unsatisfactory way, without first giving a thorough account of the general theory of optimal stopping. However, we shall give rigorous proofs relying on the method of “guess and verify”. That is to say, our proofs will start with a candidate solution, the choice of which is inspired by intuition, and then we shall prove that this candidate verifies sufficient conditions in order to confirm its status as the actual solution.

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Vorheriges Kapitel Ruin Problems and Gerber–Shiu Theory

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Right-continuity of paths is implicitly used here.

See, however, Baurdoux and van Schaik (2012) who investigate the problem of stopping as “close” the maximum as possible in an appropriate sense.

Gerber and Shiu (1994) dealt with the case of bounded variation spectrally positive Lévy processes; Boyarchenko and Levendorskii (2002a) handled a class of tempered stable processes; Chan (2004) covers the case of spectrally negative processes; Avram et al. (2002, 2004) deal with spectrally negative Lévy processes again; Asmussen et al. (2004) look at Lévy processes which have phase-type jumps and Chesney and Jeanblanc (2004) again for the spectrally negative case.

The continuous-time arguments are also given in Kyprianou and Surya (2005). Further work in this direction can be found in Deligiannidis and Utev (2009).

In fact it is the case that $\mathbb{P}(\tau^{*}<\infty) =1$ thanks to the law of the iterated logarithm for X, which states that

$$\limsup_{t\uparrow\infty}\frac{X_t}{t^{1/\alpha}(2\log\log t)^{(\alpha -1)/\alpha}} =c_\alpha $$

almost surely, for some constant c _α>0.

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Titel: Applications to Optimal Stopping Problems
verfasst von: Andreas E. Kyprianou
Verlag: Springer Berlin Heidelberg
Buch: Fluctuations of Lévy Processes with Applications
Print ISBN: 978-3-642-37631-3

Electronic ISBN: 978-3-642-37632-0

Copyright-Jahr: 2014
DOI: https://doi.org/10.1007/978-3-642-37632-0_11