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On exit times of Lévy-driven Ornstein-Uhlenbeck processes

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  • Borovkov, Konstantin
  • Novikov, Alexander

Abstract

We prove two martingale identities which involve exit times of Lévy-driven Ornstein-Uhlenbeck processes. Using these identities we find an explicit formula for the Laplace transform of the exit time under the assumption that positive jumps of the Lévy process are exponentially distributed.

Suggested Citation

  • Borovkov, Konstantin & Novikov, Alexander, 2008. "On exit times of Lévy-driven Ornstein-Uhlenbeck processes," Statistics & Probability Letters, Elsevier, vol. 78(12), pages 1517-1525, September.
    • Handle: RePEc:eee:stapro:v:78:y:2008:i:12:p:1517-1525
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    References listed on IDEAS

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    1. Borovkov, K. & Novikov, A., 2001. "On a piece-wise deterministic Markov process model," Statistics & Probability Letters, Elsevier, vol. 53(4), pages 421-428, July.
    2. Wolfe, Stephen James, 1982. "On a continuous analogue of the stochastic difference equation Xn=[rho]Xn-1+Bn," Stochastic Processes and their Applications, Elsevier, vol. 12(3), pages 301-312, May.
    3. Alexander Novikov & R. E. Melchers & E. Shinjikashvili & N. Kordzakhia, 2003. "First Passage Time of Filtered Poisson Process with Exponential Shape Function," Research Paper Series 109, Quantitative Finance Research Centre, University of Technology, Sydney.
    4. Tsurui, Akira & Osaki, Shunji, 1976. "On a first-passage problem for a cumulative process with exponential decay," Stochastic Processes and their Applications, Elsevier, vol. 4(1), pages 79-88, January.
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    Cited by:

    1. Endres, Sylvia & Stübinger, Johannes, 2017. "Optimal trading strategies for Lévy-driven Ornstein-Uhlenbeck processes," FAU Discussion Papers in Economics 17/2017, Friedrich-Alexander University Erlangen-Nuremberg, Institute for Economics.
    2. 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).
    3. Zhou, Jiang & Wu, Lan & Bai, Yang, 2017. "Occupation times of Lévy-driven Ornstein–Uhlenbeck processes with two-sided exponential jumps and applications," Statistics & Probability Letters, Elsevier, vol. 125(C), pages 80-90.
    4. Tim Leung & Kevin W. Lu, 2023. "Monte Carlo Simulation for Trading Under a L\'evy-Driven Mean-Reverting Framework," Papers 2309.05512, arXiv.org, revised Jan 2024.
    5. Shiyu Song, 2021. "Some Explicit Results on First Exit Times for a Jump Diffusion Process Involving Semimartingale Local Time," Journal of Theoretical Probability, Springer, vol. 34(4), pages 2346-2367, December.
    6. Habtemicael, Semere & SenGupta, Indranil, 2014. "Ornstein–Uhlenbeck processes for geophysical data analysis," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 399(C), pages 147-156.

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