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On the First Crossing of Two Boundaries by an Order Statistics Risk Process

Author

Listed:
  • Dimitrina S. Dimitrova

    (Faculty of Actuarial Science and Insurance, Cass Business School, City, University of London, 106 Bunhill Row, London EC1Y 8TZ, UK)

  • Zvetan G. Ignatov

    (Faculty of Economics and Business Administration, Sofia University “St Kliment Ohridski”, 125 Tsarigradsko Shosse Blv., bl.3, Sofia 1113, Bulgaria)

  • Vladimir K. Kaishev

    (Faculty of Actuarial Science and Insurance, Cass Business School, City, University of London, 106 Bunhill Row, London EC1Y 8TZ, UK)

Abstract

We derive a closed form expression for the probability that a non-decreasing, pure jump stochastic risk process with the order statistics (OS) property will not exit the strip between two non-decreasing, possibly discontinuous, time-dependent boundaries, within a finite time interval. The result yields new expressions for the ruin probability in the insurance and the dual risk models with dependence between the claim severities or capital gains respectively.

Suggested Citation

  • Dimitrina S. Dimitrova & Zvetan G. Ignatov & Vladimir K. Kaishev, 2017. "On the First Crossing of Two Boundaries by an Order Statistics Risk Process," Risks, MDPI, vol. 5(3), pages 1-14, August.
    • Handle: RePEc:gam:jrisks:v:5:y:2017:i:3:p:43-:d:108877
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    References listed on IDEAS

    as
    1. Claude Lefèvre & Philippe Picard, 2014. "Ruin Probabilities for Risk Models with Ordered Claim Arrivals," Methodology and Computing in Applied Probability, Springer, vol. 16(4), pages 885-905, December.
    2. De Vylder, F. E. & Goovaerts, M. J., 1999. "Inequality extensions of Prabhu's formula in ruin theory," Insurance: Mathematics and Economics, Elsevier, vol. 24(3), pages 249-271, May.
    3. Dimitrova, Dimitrina S. & Kaishev, Vladimir K. & Zhao, Shouqi, 2016. "On the evaluation of finite-time ruin probabilities in a dependent risk model," Applied Mathematics and Computation, Elsevier, vol. 275(C), pages 268-286.
    4. Stéphane Loisel & Claude Lefèvre, 2009. "Finite-Time Ruin Probabilities for Discrete, Possibly Dependent, Claim Severities," Post-Print hal-00201377, HAL.
    5. Claude Lefèvre & Stéphane Loisel, 2009. "Finite-Time Ruin Probabilities for Discrete, Possibly Dependent, Claim Severities," Methodology and Computing in Applied Probability, Springer, vol. 11(3), pages 425-441, September.
    6. K. Borovkov & Alexander Novikov, 2004. "Explicit Bounds for Approximation Rates for Boundary Crossing Probabilities for the Wiener Process," Research Paper Series 115, Quantitative Finance Research Centre, University of Technology, Sydney.
    7. Pierre-Olivier Goffard, 2017. "Two-sided exit problems in the ordered risk model," Working Papers hal-01528204, HAL.
    8. Lefèvre, Claude & Picard, Philippe, 2011. "A new look at the homogeneous risk model," Insurance: Mathematics and Economics, Elsevier, vol. 49(3), pages 512-519.
    9. Dimitrova, Dimitrina S. & Kaishev, Vladimir K. & Zhao, Shouqi, 2015. "On finite-time ruin probabilities in a generalized dual risk model with dependence," European Journal of Operational Research, Elsevier, vol. 242(1), pages 134-148.
    10. Ignatov, Zvetan G. & Kaishev, Vladimir K. & Krachunov, Rossen S., 2001. "An improved finite-time ruin probability formula and its Mathematica implementation," Insurance: Mathematics and Economics, Elsevier, vol. 29(3), pages 375-386, December.
    11. De Vylder, F. & Goovaerts, M., 2000. "Homogeneous risk models with equalized claim amounts," Insurance: Mathematics and Economics, Elsevier, vol. 26(2-3), pages 223-238, May.
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    Cited by:

    1. Dimitrova, Dimitrina S. & Ignatov, Zvetan G. & Kaishev, Vladimir K. & Tan, Senren, 2020. "On double-boundary non-crossing probability for a class of compound processes with applications," European Journal of Operational Research, Elsevier, vol. 282(2), pages 602-613.
    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. Pierre-O. Goffard, 2019. "Fraud risk assessment within blockchain transactions," Post-Print hal-01716687, HAL.

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