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Further Calculations for the McKean Stochastic Game for a Spectrally Negative Lévy Process: From a Point to an Interval

Part of: Game theory Stochastic processes

Published online by Cambridge University Press:  14 July 2016

E. J. Baurdoux  and
K. Van Schaik
E. J. Baurdoux*
Affiliation:
London School of Economics
K. Van Schaik*
Affiliation:
University of Bath
*
∗Postal address: Department of Statistics, London School of Economics, Houghton Street, London, WC2A 2AE, UK. Email address: e.j.baurdoux@lse.ac.uk
∗∗Current address: School of Mathematics, University of Manchester, Oxford Road, Manchester, M13 9PL, UK. Email address: kees.vanschaik@manchester.ac.uk
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Abstract

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Following Baurdoux and Kyprianou (2008) we consider the McKean stochastic game, a game version of the McKean optimal stopping problem (American put), driven by a spectrally negative Lévy process. We improve their characterisation of a saddle point for this game when the driving process has a Gaussian component and negative jumps. In particular, we show that the exercise region of the minimiser consists of a singleton when the penalty parameter is larger than some threshold and ‘thickens’ to a full interval when the penalty parameter drops below this threshold. Expressions in terms of scale functions for the general case and in terms of polynomials for a specific jump diffusion case are provided.

Keywords

Stochastic gameoptimal stoppingLévy processfluctuation theory

MSC classification

Secondary: 60G40: Stopping times; optimal stopping problems; gambling theory 91A15: Stochastic games
Type
Research Article
Information
Journal of Applied Probability , Volume 48 , Issue 1 , March 2011 , pp. 200 - 216
Copyright
Copyright © Applied Probability Trust 2011 

Footnotes

This author gratefully acknowledges being supported by a postdoctoral grant from the AXA Research Fund.

References

[1] Avram, F., Kyprianou, A. E. and Pistorius, M. R. (2004). Exit problems for spectrally negative Lévy processes and applications to (Canadized) Russian options. Ann. Appl. Prob. 14, 215–238.CrossRefGoogle Scholar
[2] Baurdoux, E. J. and Kyprianou, A. E. (2008). The McKean stochastic game driven by a spectrally negative Lévy process. Electron J. Prob. 8, 173–197.Google Scholar
[3] Baurdoux, E. J. and Kyprianou, A. E. (2009). The Shepp–Shiryaev stochastic game driven by a spectrally negative Lévy process. Theory Prob. Appl. 53, 481–499.Google Scholar
[4] Baurdoux, E. J., Kyprianou, A. E. and Pardo, J. C. (2011). The Gapeev–Kühn stochastic game driven by a spectrally positive Lévy process. To appear in Stoch. Process. Appl. Google Scholar
[5] Bertoin, J. (1996). Lévy Processes. {Cambridge University Press}.Google Scholar
[6] Bichteler, K. (2002). Stochastic Integration with Jumps. {Cambridge University Press}.Google Scholar
[7] Chan, T. (2004). Some applications of Lévy processes in insurance and finance. Finance 25, 71–94.Google Scholar
[8] Doney, R. A. (2005). Some excursion calculations for spectrally one-sided Lévy processes. In Séminaire de Probabilités XXXVIII (Lecture Notes Math. 1857), Springer, Berlin, pp. 5–15.Google Scholar
[9] Dynkin, E. B. (1969). A game-theoretic version of an optimal stopping problem. Dokl. Akad. Nauk. SSSR 185, 16–19.Google Scholar
[10] Ekström, E. and Peskir, G. (2008). Optimal stopping games for Markov processes. SIAM J. Control Optimization 2, 684–702.Google Scholar
[11] Gapeev, P. V. and Kühn, C. (2005). Perpetual convertible bonds in Jump-diffusion models. Statist. Decisions 23, 15–31.Google Scholar
[12] Kallsen, J. and Kühn, C. (2004). Pricing derivatives of American and game type in incomplete markets. Finance Stoch. 8, 261–284.CrossRefGoogle Scholar
[13] Kifer, Y. (2000). Game options. Finance Stoch. 4, 443–463.Google Scholar
[14] Kou, S. and Wang, H. (2002). Option pricing under a Jump-diffusion model. Manag. Sci. 50, 1178–1192.Google Scholar
[15] Kyprianou, A. E. (2004). Some calculations for Israeli options. Finance Stoch. 8, 73–86.Google Scholar
[16] Kyprianou, A. E. (2006). Introductory Lectures on Fluctuations of Lévy Processes with Applications. Springer, Berlin.Google Scholar
[17] McKean, H. (1965). Appendix: a free boundary problem for the heat equation arising from a problem of mathematical economics. Indust. Manag. Rev. 6, 32–39.Google Scholar
[18] Mordecki, E. (2002). Optimal stopping and perpetual options for Lévy processes. Finance Stoch. 6, 473–493.Google Scholar