Modelling Extremal Events pp 21-57

Part of the Applications of Mathematics book series (SMAP, volume 33)

Risk Theory

  • Paul Emberchts
  • Claudia Klüppelberg
  • Thomas Mikosch


For most of the problems treated in insurance mathematics, risk theory still provides the quintessential mathematical basis. The present chapter will serve a similar purpose for the rest of this book. The basic risk theory models will be introduced, stressing the instances where a division between small and large claims is relevant. Nowadays, there is a multitude of textbooks available treating risk theory at various mathematical levels. Consequently, our treatment will not be encyclopaedic, but will focus more on those aspects of the theory where we feel that, for modelling extremal events, the existing literature needs complementing. Readers with a background in finance rather than insurance may use this chapter as a first introduction to the stochastic modelling of claim processes.


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Notes and Comments

  1. 630.
    Veraverbeke, N. (1993) Asymptotic estimates for the probability of ruin in a Poisson model with diffusion. Insurance: Math. Econom. 13, 57–62. [56] Google Scholar
  2. 248.
    Furrer, H.J. and Schmidli, H. (1994) Exponential inequalities for ruin probabilities of risk processes perturbed by diffusion. Insurance: Math. Econom.15, 23–36. [56] Google Scholar
  3. 31.
    Asmussen, S., Floe Henriksen, L. and Klüppelberg, C. (1994) Large claims approximations for risk processes in a Markovian environment. Stoch. Proc. Appl. 54, 29–43. [56, 454] Google Scholar
  4. 524.
    Reinhard, J.M. (1984) On a class of semi-Markov risk models obtained as classical risk models in a Markovian environment. ASTIN Bulletin 14, 2343. [57] Google Scholar
  5. 399.
    Klüppelberg, C. and Stadtmüller, U. (1998) Ruin probabilities in the presence of heavy-tails and interest rates. Scand. Actuar. J., 49–58. [57] Google Scholar
  6. 29.
    Asmussen, S. (1998) Subexponential asymptotics for stochastic processes: extremal behaviour, stationary distributions and first passage probabilities. Ann. Appl. Probab. 8, 354–374. [57] Google Scholar
  7. 201.
    Embrechts, P. (1983) A property of the generalized inverse Gaussian distribution with some applications. J. Appl. Probab. 20, 537–544. [36, 57] Google Scholar
  8. 486.
    Omey, E. and Willekens, E. (1986) Second order behaviour of the tail of a subordinated probability distribution. Stoch. Proc. Appl. 21, 339–353. [57] Google Scholar
  9. 487.
    Omey, E. and Willekens, E. (1987) Second order behaviour of distributions subordinate to a distribution with finite mean. Commun. Statist. Stochastic Models 3, 311–342. [57] MathSciNetGoogle Scholar
  10. 30.
    Asmussen, S. and Binswanger, K. (1997) Simulation of ruin probabilities for subexponential claims. ASTIN Bulletin 27, 297–318. [57, 454] Google Scholar
  11. 74.
    Binswanger, K. (1997) Rare Events in Insurance. PhD thesis, ETH Zürich. [57] Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1997

Authors and Affiliations

  • Paul Emberchts
    • 1
  • Claudia Klüppelberg
    • 2
  • Thomas Mikosch
    • 3
  1. 1.Department of MathematicsETH ZurichZurichSwitzerland
  2. 2.Center for Mathematical SciencesMunich University of TechnologyGarchingGermany
  3. 3.Laboratory of Actuarial MathematicsUniversity of CopenhagenCopenhagenDenmark

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