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A Two-Sided First-Exit Problem for a Compound Poisson Process with a Random Upper Boundary

Author

Listed:
  • D. Perry

    (University of Haifa)

  • W. Stadje

    (University of Osnabrück)

  • S. Zacks

    (Binghamton University)

Abstract

We consider the first-exit time of a compound Poisson process from a region that is bounded from below by an increasing straight line, while its upper boundary has positive jumps of i.i.d. sizes at Poisson times and increases linearly between jumps. An integral equation for the corresponding Laplace-Stieltjes transforms is derived and solved. The case of exponential jumps is treated separately. The problem has applications in queueing and risk theory.

Suggested Citation

  • D. Perry & W. Stadje & S. Zacks, 2005. "A Two-Sided First-Exit Problem for a Compound Poisson Process with a Random Upper Boundary," Methodology and Computing in Applied Probability, Springer, vol. 7(1), pages 51-62, March.
    • Handle: RePEc:spr:metcap:v:7:y:2005:i:1:d:10.1007_s11009-005-6654-6
    DOI: 10.1007/s11009-005-6654-6
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    References listed on IDEAS

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    1. Gerber, Hans U., 1988. "Mathematical fun with ruin theory," Insurance: Mathematics and Economics, Elsevier, vol. 7(1), pages 15-23, January.
    2. Picard, Philippe & Lefevre, Claude, 1994. "On the first crossing of the surplus process with a given upper barrier," Insurance: Mathematics and Economics, Elsevier, vol. 14(2), pages 163-179, May.
    3. Picard, Philippe & Lefèvre, Claude, 2003. "On the first meeting or crossing of two independent trajectories for some counting processes," Stochastic Processes and their Applications, Elsevier, vol. 104(2), pages 217-242, April.
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    Cited by:

    1. Pierre-Olivier Goffard, 2019. "Two-sided exit problems in the ordered risk model," Post-Print hal-01528204, HAL.
    2. Pierre-Olivier Goffard, 2019. "Two-Sided Exit Problems in the Ordered Risk Model," Methodology and Computing in Applied Probability, Springer, vol. 21(2), pages 539-549, June.
    3. Pierre-Olivier Goffard, 2019. "Fraud risk assessment within blockchain transactions," Working Papers hal-01716687, HAL.
    4. Antonio Di Crescenzo & Barbara Martinucci, 2013. "On the Generalized Telegraph Process with Deterministic Jumps," Methodology and Computing in Applied Probability, Springer, vol. 15(1), pages 215-235, March.
    5. Shuanming Li & Yi Lu & Can Jin, 2016. "Number of Jumps in Two-Sided First-Exit Problems for a Compound Poisson Process," Methodology and Computing in Applied Probability, Springer, vol. 18(3), pages 747-764, September.
    6. Li, Shuanming & Lu, Yi, 2017. "Distributional study of finite-time ruin related problems for the classical risk model," Applied Mathematics and Computation, Elsevier, vol. 315(C), pages 319-330.
    7. Pierre-O. Goffard, 2019. "Fraud risk assessment within blockchain transactions," Post-Print hal-01716687, HAL.

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