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On the optimality of threshold type strategies in single and recursive optimal stopping under Lévy models

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  • Long, Mingsi
  • Zhang, Hongzhong

Abstract

In the spirit of Surya (2007), we develop an average problem approach to prove the optimality of threshold type strategies for optimal stopping of Lévy models with a continuous additive functional (CAF) discounting. Under spectrally negative models, we specialize this in terms of conditions on the reward function and random discounting, where we present two examples of local time and occupation time discounting. We then apply this approach to recursive optimal stopping problems, and present simpler and neater proofs for a number of important results on qualitative properties of the optimal thresholds, which are only known under a few special cases Carmona and Touzi (2008), Leung et al. (2015) and Surya (2007).

Suggested Citation

  • Long, Mingsi & Zhang, Hongzhong, 2019. "On the optimality of threshold type strategies in single and recursive optimal stopping under Lévy models," Stochastic Processes and their Applications, Elsevier, vol. 129(8), pages 2821-2849.
    • Handle: RePEc:eee:spapps:v:129:y:2019:i:8:p:2821-2849
    DOI: 10.1016/j.spa.2018.08.005
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    References listed on IDEAS

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    1. Tim Leung & Kazutoshi Yamazaki & Hongzhong Zhang, 2015. "An Analytic Recursive Method For Optimal Multiple Stopping: Canadization And Phase-Type Fitting," International Journal of Theoretical and Applied Finance (IJTAF), World Scientific Publishing Co. Pte. Ltd., vol. 18(05), pages 1-31.
    2. René Carmona & Savas Dayanik, 2008. "Optimal Multiple Stopping of Linear Diffusions," Mathematics of Operations Research, INFORMS, vol. 33(2), pages 446-460, May.
    3. Savas Dayanik, 2008. "Optimal Stopping of Linear Diffusions with Random Discounting," Mathematics of Operations Research, INFORMS, vol. 33(3), pages 645-661, August.
    4. Amina Bouzguenda Zeghal & Mohamed Mnif, 2006. "Optimal Multiple Stopping And Valuation Of Swing Options In Lévy Models," International Journal of Theoretical and Applied Finance (IJTAF), World Scientific Publishing Co. Pte. Ltd., vol. 9(08), pages 1267-1297.
    5. René Carmona & Nizar Touzi, 2008. "Optimal Multiple Stopping And Valuation Of Swing Options," Mathematical Finance, Wiley Blackwell, vol. 18(2), pages 239-268, April.
    6. Tim Leung & Kazutoshi Yamazaki & Hongzhong Zhang, 2015. "Optimal Multiple Stopping with Negative Discount Rate and Random Refraction Times under Levy Models," Papers 1505.07313, arXiv.org.
    7. 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.
    8. 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.
    9. Loeffen, Ronnie L. & Renaud, Jean-François & Zhou, Xiaowen, 2014. "Occupation times of intervals until first passage times for spectrally negative Lévy processes," Stochastic Processes and their Applications, Elsevier, vol. 124(3), pages 1408-1435.
    10. Ernesto Mordecki, 2002. "Optimal stopping and perpetual options for Lévy processes," Finance and Stochastics, Springer, vol. 6(4), pages 473-493.
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    Cited by:

    1. Neofytos Rodosthenous & Hongzhong Zhang, 2020. "When to sell an asset amid anxiety about drawdowns," Mathematical Finance, Wiley Blackwell, vol. 30(4), pages 1422-1460, October.
    2. Zbigniew Palmowski & José Luis Pérez & Budhi Arta Surya & Kazutoshi Yamazaki, 2020. "The Leland–Toft optimal capital structure model under Poisson observations," Finance and Stochastics, Springer, vol. 24(4), pages 1035-1082, October.
    3. Neofytos Rodosthenous & Hongzhong Zhang, 2020. "When to sell an asset amid anxiety about drawdowns," Papers 2006.00282, arXiv.org.

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