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A note on one-sided solutions for optimal stopping problems driven by Lévy processes

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  • Lin, Yi-Shen

Abstract

All optimal stopping problems with increasing, logconcave and right-continuous reward functions under Lévy processes have been shown to admit one-sided solutions. There is relatively little research attention, however, on constructing an explicit solution of the problem when the reward function is increasing and logconcave. Thus, this paper aims to explore the role of the monotonicity and logconcavity in the characterization of the optimal threshold value, particularly in terms of the ladder height process. In this paper, with additional smoothness assumptions on the reward function, we give an explicit expression for the optimal threshold value provided that the underlying Lévy process drifts to −∞ almost surely. In the particular case where the reward function is (x+)ν=(max{x,0})ν, ν∈(0,∞), we establish a connection indicating that the expression coincides with the positive root of the associated Appell function.

Suggested Citation

  • Lin, Yi-Shen, 2024. "A note on one-sided solutions for optimal stopping problems driven by Lévy processes," Statistics & Probability Letters, Elsevier, vol. 206(C).
  • Handle: RePEc:eee:stapro:v:206:y:2024:i:c:s0167715223002134
    DOI: 10.1016/j.spl.2023.109989
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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.
    2. Ernesto Mordecki, 2002. "Optimal stopping and perpetual options for Lévy processes," Finance and Stochastics, Springer, vol. 6(4), pages 473-493.
    3. Christensen, Sören & Salminen, Paavo & Ta, Bao Quoc, 2013. "Optimal stopping of strong Markov processes," Stochastic Processes and their Applications, Elsevier, vol. 123(3), pages 1138-1159.
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