The structure of the stopping region in a Lévy model

Authors:
A. G. Moroz and G. M. Shevchenko

Translated by:
S. V. Kvasko

Original publication:
Teoriya Imovirnostei ta Matematichna Statistika, tom **84** (2011).

Journal:
Theor. Probability and Math. Statist. **84** (2012), 107-115

MSC (2010):
Primary 60G40, 60G51

Published electronically:
July 31, 2012

MathSciNet review:
2857421

Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: The optimal stopping problem in a Lévy model is investigated. We show that the stopping region is nonempty for a wide class of models and payoff functions. In the general case, we establish sufficient conditions on the payoff function that provide nonemptiness of the stopping region. For a zero discounting rate we also give conditions for the stopping region to have a threshold structure.

**1.**S. Villeneuve,*Exercise regions of American options on several assets*, Finance Stoch.**3**(1999), no. 3, 295-322.**2.**Damien Lamberton and Mohammed Mikou,*The critical price for the American put in an exponential Lévy model*, Finance Stoch.**12**(2008), no. 4, 561–581. MR**2447412**, 10.1007/s00780-008-0073-9**3.**A. G. Kukush, Yu. S. Mishura, and G. M. Shevchenko,*On reselling of European option*, Theory Stoch. Process.**12**(2006), no. 3-4, 75–87. MR**2316567****4.**A. Moroz and G. Shevchenko,*Asymptotic behavior of the American type option prices in the Lévy model if the time interval is extending unboundedly*, Visn. Kyiv Univ. Mat. Mech.**24**(2010), 39-43. (Ukrainian)**5.**H. Jönsson, A. G. Kukush, and D. S. Silvestrov,*Threshold structure of optimal stopping strategies for American type option. I*, Teor. Ĭmovīr. Mat. Stat.**71**(2004), 82–92; English transl., Theory Probab. Math. Statist.**71**(2005), 93–103. MR**2144323**, 10.1090/S0094-9000-06-00650-8**6.**H. Jönsson, A. G. Kukush, and D. S. Silvestrov,*Threshold structure of optimal stopping strategies for American type option. II*, Teor. Ĭmovīr. Mat. Stat.**72**(2005), 42–53; English transl., Theory Probab. Math. Statist.**72**(2006), 47–58. MR**2168135**, 10.1090/S0094-9000-06-00663-6**7.**A. Papapantoleon,*An Introduction to Lévy Processes with Applications in Finance*, Lecture notes, 2008; arXiv/0804.0482.**8.**Philip E. Protter,*Stochastic integration and differential equations*, 2nd ed., Applications of Mathematics (New York), vol. 21, Springer-Verlag, Berlin, 2004. Stochastic Modelling and Applied Probability. MR**2020294**

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Additional Information

**A. G. Moroz**

Affiliation:
Department of Probability Theory, Statistics, and Actuarial Mathematics, Faculty for Mechanics and Mathematics, National Taras Shevchenko University, Academician Glushkov Avenue 4E, Kiev 03127, Ukraine

Email:
mag-87@inbox.ru

**G. M. Shevchenko**

Affiliation:
Department of Probability Theory, Statistics, and Actuarial Mathematics, Faculty for Mechanics and Mathematics, National Taras Shevchenko University, Academician Glushkov Avenue 4E, Kiev 03127, Ukraine

Email:
zhora@univ.kiev.ua

DOI:
http://dx.doi.org/10.1090/S0094-9000-2012-00864-8

Keywords:
Lévy processes,
American option,
payoff function,
stopping region,
threshold structure

Received by editor(s):
April 11, 2011

Published electronically:
July 31, 2012

Article copyright:
© Copyright 2012
American Mathematical Society