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The Gapeev-Kühn stochastic game driven by a spectrally positive Lévy process

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  • Baurdoux, E.J.
  • Kyprianou, A.E.
  • Pardo, J.C.

Abstract

In Gapeev and Kühn (2005) [8], the Dynkin game corresponding to perpetual convertible bonds was considered, when driven by a Brownian motion and a compound Poisson process with exponential jumps. We consider the same stochastic game but driven by a spectrally positive Lévy process. We establish a complete solution to the game indicating four principle parameter regimes as well as characterizing the occurrence of continuous and smooth fit. In Gapeev and Kühn (2005) [8], the method of proof was mainly based on solving a free boundary value problem. In this paper, we instead use fluctuation theory and an auxiliary optimal stopping problem to find a solution to the game.

Suggested Citation

  • Baurdoux, E.J. & Kyprianou, A.E. & Pardo, J.C., 2011. "The Gapeev-Kühn stochastic game driven by a spectrally positive Lévy process," Stochastic Processes and their Applications, Elsevier, vol. 121(6), pages 1266-1289, June.
    • Handle: RePEc:eee:spapps:v:121:y:2011:i:6:p:1266-1289
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    References listed on IDEAS

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    1. L. Alili & A. E. Kyprianou, 2005. "Some remarks on first passage of Levy processes, the American put and pasting principles," Papers math/0508487, arXiv.org.
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    Cited by:

    1. Egami, Masahiko & Leung, Tim & Yamazaki, Kazutoshi, 2013. "Default swap games driven by spectrally negative Lévy processes," Stochastic Processes and their Applications, Elsevier, vol. 123(2), pages 347-384.
    2. Hernández-Hernández, Daniel & Yamazaki, Kazutoshi, 2015. "Games of singular control and stopping driven by spectrally one-sided Lévy processes," Stochastic Processes and their Applications, Elsevier, vol. 125(1), pages 1-38.
    3. Gapeev, Pavel V. & Rodosthenous, Neofytos, 2016. "Perpetual American options in diffusion-type models with running maxima and drawdowns," Stochastic Processes and their Applications, Elsevier, vol. 126(7), pages 2038-2061.

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