Electronic Journal of Probability

The McKean stochastic game driven by a spectrally negative Lévy process

Erik Baurdoux and Andreas Kyprianou

Full-text: Open access

Abstract

We consider the stochastic-game-analogue of McKean's optimal stopping problem when the underlying source of randomness is a spectrally negative Lévy process. Compared to the solution for linear Brownian motion given in Kyprianou (2004) one finds two new phenomena. Firstly the breakdown of smooth fit and secondly the stopping domain for one of the players `thickens' from a singleton to an interval, at least in the case that there is no Gaussian component.

Article information

Source
Electron. J. Probab., Volume 13 (2008), paper no. 8, 173-197.

Dates
Accepted: 14 February 2008
First available in Project Euclid: 1 June 2016

Permanent link to this document
https://projecteuclid.org/euclid.ejp/1464819081

Digital Object Identifier
doi:10.1214/EJP.v13-484

Mathematical Reviews number (MathSciNet)
MR2386731

Zentralblatt MATH identifier
1190.60084

Subjects
Primary: 60J99: None of the above, but in this section
Secondary: 60G40: Stopping times; optimal stopping problems; gambling theory [See also 62L15, 91A60] 91B70: Stochastic models

Keywords
Stochastic games optimal stopping pasting principles fluctuation theory L'evy processes

Rights
This work is licensed under aCreative Commons Attribution 3.0 License.

Citation

Baurdoux, Erik; Kyprianou, Andreas. The McKean stochastic game driven by a spectrally negative Lévy process. Electron. J. Probab. 13 (2008), paper no. 8, 173--197. doi:10.1214/EJP.v13-484. https://projecteuclid.org/euclid.ejp/1464819081


Export citation

References

  • Alili, Larbi; Kyprianou, Andreas E. Some remarks on first passage of Lévy processes, the American put and pasting principles. Ann. Appl. Probab. 15 (2005), no. 3, 2062–2080.
  • Avram, Florin; Kyprianou, Andreas E.; Pistorius, Martijn R. Exit problems for spectrally negative Lévy processes and applications to (Canadized) Russian options. Ann. Appl. Probab. 14 (2004), no. 1, 215–238.
  • Baurdoux, Erik J. Fluctuation Theory and Stochastic Games for Spectrally Negative Lévy Processes. Ph.D. thesis, Utrecht University (2007).
  • Baurdoux, Erik J.; Kyprianou, A.E. The Shepp–Shiryaev stochastic game driven by a spectrally negative Lévy process. Preprint (2008).
  • Bertoin, Jean. Lévy processes. Cambridge Tracts in Mathematics, 121. Cambridge University Press, Cambridge, 1996. x+265 pp. ISBN: 0-521-56243-0
  • Bichteler, Klaus. Stochastic integration with jumps. Encyclopedia of Mathematics and its Applications, 89. Cambridge University Press, Cambridge, 2002. xiv+501 pp. ISBN: 0-521-81129-5
  • Boyarchenko, Svetlana I.; Levendorskii, Sergei. Z. Perpetual American options under Lévy processes. SIAM J. Control Optim. 40 (2002), no. 6, 1663–1696 (electronic).
  • Chan, Terence. Some applications of Lévy processes in insurance and finance. Finance. (2004) 25, 71–94.
  • Cvitanic, Jaksa; Karatzas, Ioannis. Backward stochastic differential equations with reflection and Dynkin games. Ann. Probab. 24 (1996), no. 4, 2024–2056.
  • Doney, Ronald A. Some excursion calculations for spectrally one-sided Lévy processes. Séminaire de Probabilités XXXVIII, 5–15, Lecture Notes in Math., 1857, Springer, Berlin, 2005.
  • Dynkin, Eugene. B. A game-theoretic version of an optimal stopping problem. (Russian) Dokl. Akad. Nauk SSSR 185 (1969) 16–19.
  • Ekström, Erik; Peskir Goran. Optimal stopping games for Markov processes. To appear SIAM J. Control Optim. (2006).
  • Gapeev, Pavel. V. Problems of the sequential discrimination of hypotheses for a compound Poisson process with exponential jumps. (Russian) Uspekhi Mat. Nauk 57 (2002), no. 6(348), 171–172; translation in Russian Math. Surveys 57 (2002), no. 6, 1220–1221
  • Gapeev, Pavel V.; Kühn, Christoph. Perpetual convertible bonds in jump-diffusion models. Statist. Decisions 23 (2005), no. 1, 15–31.
  • Kifer, Yuri. Game options. Finance Stoch. 4 (2000), no. 4, 443–463.
  • Kyprianou, Andreas E. Some calculations for Israeli options. Finance Stoch. 8 (2004), no. 1, 73–86.
  • Kyprianou, Andreas E. Introductory lectures on fluctuations of Lévy processes with applications. Universitext. Springer-Verlag, Berlin, 2006. xiv+373 pp. ISBN: 978-3-540-31342-7; 3-540-31342-7 r.
  • Kyprianou, Andreas E.; Surya, Budhi A. A note on a change of variable formula with local time-space for Lévy processes of bounded variation Séminaire de Probabilités XL 97–105, Lecture Notes in Math., 1899, Springer, Berlin, 2007.
  • Kyprianou, Andreas E., Rivero, Victor; Song, Renming. Smoothness and convexity of scale functions with applications to de Finetti's control problem. http://arxiv.org/abs/0801.1951.
  • McKean, Henry. Appendix: A free boundary problem for the heat equation arising from a problem of mathematical economics. Ind. Manag. Rev. 6 (1965), 32–39.
  • Millar, P. Warwick. Zero-one laws and the minimum of a Markov process. Trans. Amer. Math. Soc. 226 (1977), 365–391.
  • Mordecki, Ernesto. Optimal stopping and perpetual options for Lévy processes. Finance Stoch. 6 (2002), no. 4, 473–493.
  • Peskir, Goran.; Shiryaev, Albert. N. Sequential testing problems for Poisson processes. Ann. Statist. 28 (2000), no. 3, 837–859.
  • Peskir, Goran; Shiryaev, Albert N. Solving the Poisson disorder problem. Advances in finance and stochastics, 295–312, Springer, Berlin, 2002.
  • Peskir, Goran; Shiryaev, Albert N. Optimal stopping and free-boundary problems. Lectures in Mathematics ETH Zürich. Birkhäuser Verlag, Basel, 2006. xxii+500 pp. ISBN: 978-3-7643-2419-3; 3-7643-2419-8 (Review)r.
  • Pistorius, Martijn R. A potential-theoretical review of some exit problems of spectrally negative Lévy processes. Séminaire de Probabilités XXXVIII, 30–41, Lecture Notes in Math., 1857, Springer, Berlin, 2005.
  • Protter, Philip E. Stochastic integration and differential equations. Second edition. Applications of Mathematics (New York), 21. Stochastic Modelling and Applied Probability. Springer-Verlag, Berlin, 2004. xiv+415 pp. ISBN: 3-540-00313-4
  • Shiryaev, Albert, N. Optimal stopping rules. Translated from the Russian by A. B. Aries. Applications of Mathematics, Vol. 8. Springer-Verlag, New York-Heidelberg, 1978. x+217 pp. ISBN: 0-387-90256-2 62L15 (60G40)
  • Surya, Budhi A. An approach for solving perpetual optimal stopping problems driven by Lévy processes. Stochastics 79 (2007), no. 3-4, 337–361.