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Large deviations results for subexponential tails, with applications to insurance risk

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  • Asmussen, Søren
  • Klüppelberg, Claudia

Abstract

Consider a random walk or Lévy process {St} and let [tau](u) = inf {t[greater-or-equal, slanted]0 : St > u}, P(u)(·) = P(· [tau](u) [infinity]. Essentially, the results confirm the folklore that level crossing occurs as result of one big jump. Particular sharp conclusions are obtained for downwards skip-free processes like the classical compound Poisson insurance risk process where the formulation is in terms of total variation convergence. The ideas of the proof involve excursions and path decompositions for Markov processes. As a corollary, it follows that for some deterministic function a(u), the limiting P(u)-distribution of [tau](u)/a(u) is either Pareto or exponential, and corresponding approximations for the finite time ruin probabilities are given.

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  • Asmussen, Søren & Klüppelberg, Claudia, 1996. "Large deviations results for subexponential tails, with applications to insurance risk," Stochastic Processes and their Applications, Elsevier, vol. 64(1), pages 103-125, November.
    • Handle: RePEc:eee:spapps:v:64:y:1996:i:1:p:103-125
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    References listed on IDEAS

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    1. Asmussen, Søren & Henriksen, Lotte Fløe & Klüppelberg, Claudia, 1994. "Large claims approximations for risk processes in a Markovian environment," Stochastic Processes and their Applications, Elsevier, vol. 54(1), pages 29-43, November.
    2. Asmussen, Søren & Schmidt, Volker, 1995. "Ladder height distributions with marks," Stochastic Processes and their Applications, Elsevier, vol. 58(1), pages 105-119, July.
    3. Dufresne, Francois & Gerber, Hans U., 1988. "The surpluses immediately before and at ruin, and the amount of the claim causing ruin," Insurance: Mathematics and Economics, Elsevier, vol. 7(3), pages 193-199, October.
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    5. Embrechts, P. & Veraverbeke, N., 1982. "Estimates for the probability of ruin with special emphasis on the possibility of large claims," Insurance: Mathematics and Economics, Elsevier, vol. 1(1), pages 55-72, January.
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    Cited by:

    1. Zhu, Lingjiong, 2013. "Ruin probabilities for risk processes with non-stationary arrivals and subexponential claims," Insurance: Mathematics and Economics, Elsevier, vol. 53(3), pages 544-550.
    2. Harri Nyrhinen, 2009. "On Large Deviations of Multivariate Heavy-Tailed Random Walks," Journal of Theoretical Probability, Springer, vol. 22(1), pages 1-17, March.
    3. Serguei Foss & Andrew Richards, 2010. "On Sums of Conditionally Independent Subexponential Random Variables," Mathematics of Operations Research, INFORMS, vol. 35(1), pages 102-119, February.
    4. Hägele, Miriam, 2020. "Precise asymptotics of ruin probabilities for a class of multivariate heavy-tailed distributions," Statistics & Probability Letters, Elsevier, vol. 166(C).
    5. Grandell, Jan, 2000. "Simple approximations of ruin probabilities," Insurance: Mathematics and Economics, Elsevier, vol. 26(2-3), pages 157-173, May.
    6. Kamphorst, Bart & Zwart, Bert, 2019. "Uniform asymptotics for compound Poisson processes with regularly varying jumps and vanishing drift," Stochastic Processes and their Applications, Elsevier, vol. 129(2), pages 572-603.
    7. Lehtomaa, Jaakko, 2015. "Limiting behaviour of constrained sums of two variables and the principle of a single big jump," Statistics & Probability Letters, Elsevier, vol. 107(C), pages 157-163.
    8. Wolfgang Stadje, 1998. "Level-Crossing Properties of the Risk Process," Mathematics of Operations Research, INFORMS, vol. 23(3), pages 576-584, August.
    9. Julien Trufin & Stéphane Loisel, 2013. "Ultimate ruin probability in discrete time with Bühlmann credibility premium adjustments," Post-Print hal-00426790, HAL.
    10. Huyen Pham, 2007. "Some applications and methods of large deviations in finance and insurance," Papers math/0702473, arXiv.org, revised Feb 2007.
    11. Søren Asmussen & Romain Biard, 2011. "Ruin probabilities for a regenerative Poisson gap generated risk process," Post-Print hal-00569254, HAL.
    12. Sem Borst & Bert Zwart, 2005. "Fluid Queues with Heavy-Tailed M/G/∞ Input," Mathematics of Operations Research, INFORMS, vol. 30(4), pages 852-879, November.
    13. Schmidli, Hanspeter, 2010. "On the Gerber-Shiu function and change of measure," Insurance: Mathematics and Economics, Elsevier, vol. 46(1), pages 3-11, February.
    14. Harri Nyrhinen, 2015. "On real growth and run-off companies in insurance ruin theory," Papers 1511.01763, arXiv.org.
    15. Gao, Fuqing & Yan, Jun, 2009. "Sample path large and moderate deviations for risk model with delayed claims," Insurance: Mathematics and Economics, Elsevier, vol. 45(1), pages 74-80, August.
    16. Nam Kyoo Boots & Perwez Shahabuddin, 2001. "Simulating Tail Probabilities in GI/GI.1 Queues and Insurance Risk Processes with Subexponentail Distributions," Tinbergen Institute Discussion Papers 01-012/4, Tinbergen Institute.
    17. Korshunov, Dmitry, 2018. "On subexponential tails for the maxima of negatively driven compound renewal and Lévy processes," Stochastic Processes and their Applications, Elsevier, vol. 128(4), pages 1316-1332.
    18. Gyllenberg, Mats & S. Silvestrov, Dmitrii, 2000. "Cramer-Lundberg approximation for nonlinearly perturbed risk processes," Insurance: Mathematics and Economics, Elsevier, vol. 26(1), pages 75-90, February.
    19. Tang, Qihe, 2007. "The overshoot of a random walk with negative drift," Statistics & Probability Letters, Elsevier, vol. 77(2), pages 158-165, January.
    20. Griffin, Philip S. & Maller, Ross A. & Roberts, Dale, 2013. "Finite time ruin probabilities for tempered stable insurance risk processes," Insurance: Mathematics and Economics, Elsevier, vol. 53(2), pages 478-489.
    21. Wang, Kaiyong & Yang, Yang & Yu, Changjun, 2013. "Estimates for the overshoot of a random walk with negative drift and non-convolution equivalent increments," Statistics & Probability Letters, Elsevier, vol. 83(6), pages 1504-1512.
    22. Schmidli, Hanspeter, 2010. "Conditional law of risk processes given that ruin occurs," Insurance: Mathematics and Economics, Elsevier, vol. 46(2), pages 281-289, April.

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