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A Unified Approach for Drawdown (Drawup) of Time-Homogeneous Markov Processes

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  • David Landriault
  • Bin Li
  • Hongzhong Zhang

Abstract

Drawdown (resp. drawup) of a stochastic process, also referred as the reflected process at its supremum (resp. infimum), has wide applications in many areas including financial risk management, actuarial mathematics and statistics. In this paper, for general time-homogeneous Markov processes, we study the joint law of the first passage time of the drawdown (resp. drawup) process, its overshoot, and the maximum of the underlying process at this first passage time. By using short-time pathwise analysis, under some mild regularity conditions, the joint law of the three drawdown quantities is shown to be the unique solution to an integral equation which is expressed in terms of fundamental two-sided exit quantities of the underlying process. Explicit forms for this joint law are found when the Markov process has only one-sided jumps or is a L\'{e}vy process (possibly with two-sided jumps). The proposed methodology provides a unified approach to study various drawdown quantities for the general class of time-homogeneous Markov processes.

Suggested Citation

  • David Landriault & Bin Li & Hongzhong Zhang, 2017. "A Unified Approach for Drawdown (Drawup) of Time-Homogeneous Markov Processes," Papers 1702.07786, arXiv.org.
    • Handle: RePEc:arx:papers:1702.07786
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    Cited by:

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    2. Zhang, Gongqiu & Li, Lingfei, 2023. "A general method for analysis and valuation of drawdown risk," Journal of Economic Dynamics and Control, Elsevier, vol. 152(C).
    3. Florin Avram & Jose-Luis Perez-Garmendia, 2019. "A Review of First-Passage Theory for the Segerdahl-Tichy Risk Process and Open Problems," Risks, MDPI, vol. 7(4), pages 1-21, November.
    4. Li, Shu & Zhou, Xiaowen, 2022. "The Parisian and ultimate drawdowns of Lévy insurance models," Insurance: Mathematics and Economics, Elsevier, vol. 107(C), pages 140-160.
    5. Zhang, Xiang & Li, Lingfei & Zhang, Gongqiu, 2021. "Pricing American drawdown options under Markov models," European Journal of Operational Research, Elsevier, vol. 293(3), pages 1188-1205.
    6. He Yi & Lirong Cui & Narayanaswamy Balakrishnan, 2022. "On the Derivative Counting Processes of First- and Second-order Aggregated Semi-Markov Systems," Methodology and Computing in Applied Probability, Springer, vol. 24(3), pages 1849-1875, September.
    7. Foucart, Clément & Vidmar, Matija, 2024. "Continuous-state branching processes with collisions: First passage times and duality," Stochastic Processes and their Applications, Elsevier, vol. 167(C).
    8. Wenyuan Wang & Zhimin Zhang, 2019. "Optimal loss-carry-forward taxation for L\'{e}vy risk processes stopped at general draw-down time," Papers 1904.08029, arXiv.org.
    9. Vidmar, Matija, 2023. "Complete monotonicity of time-changed Lévy processes at first passage," Statistics & Probability Letters, Elsevier, vol. 193(C).

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