Advertisement

The Gerber–Shiu Measure

  • Andreas E. Kyprianou
Part of the EAA Series book series (EAAS)

Abstract

Having introduced scale functions, we look at the Gerber–Shiu measure, which characterises the joint law of the discounted wealth prior to ruin and deficit at ruin. We shall develop an idea from Chapter  4, involving Bernoulli trials of excursions from the minimum, to provide an identity for the expected occupation measure until ruin of the Cramér–Lundberg process. This identity will then play a key role in identifying an expression for the Gerber–Shiu measure.

References

  1. Bertoin, J.: Lévy Processes. Cambridge University Press, Cambridge (1996) zbMATHGoogle Scholar
  2. Bertoin, J.: Exponential decay and ergodicity of completely asymmetric Lévy processes in a finite interval. Ann. Appl. Probab. 7, 156–169 (1997) MathSciNetCrossRefzbMATHGoogle Scholar
  3. Biffis, E., Morales, M.: On a generalization of the Gerber–Shiu function to path-dependent penalties. Insurance Math. Econom. 46, 92–97 (2010) MathSciNetCrossRefzbMATHGoogle Scholar
  4. Bingham, N.H.: Fluctuation theory in continuous time. Adv. Appl. Probab. 7, 705–766 (1975) MathSciNetCrossRefzbMATHGoogle Scholar
  5. Feller, W.: An Introduction to Probability Theory and Its Applications, vol II, 2nd edn. Wiley, New York (1971) zbMATHGoogle Scholar
  6. Itô, K.: Poisson point processes attached to Markov processes. In: Proceedings of the Sixth Berkeley Symposium on Mathematical Statistics and Probability (Univ. California, Berkeley, Calif., 1970/1971), Vol. III: Probability Theory, pp. 225–239. University of California Press, Berkeley (1972) Google Scholar
  7. Korolyuk, V.S.: Boundary problems for a compound Poisson process. Theory Probab. Appl. 19, 1–14 (1974) CrossRefzbMATHGoogle Scholar
  8. Korolyuk, V.S.: Boundary Problems for Compound Poisson Processes. Naukova Dumka, Kiev (1975a) Google Scholar
  9. Korolyuk, V.S.: On ruin problem for compound Poisson process. Theory Probab. Appl. 20, 374–376 (1975b) CrossRefzbMATHGoogle Scholar
  10. Korolyuk, V.S., Borovskich, Ju.V.: Analytic Problems of the Asymptotic Behaviour of Probability Distributions. Naukova Dumka, Kiev (1981) Google Scholar
  11. Korolyuk, V.S., Suprun, V.N., Shurenkov, V.M.: Method of potential in boundary problems for processes with independent increments and jumps of the same sign. Theory Probab. Appl. 21, 243–249 (1976) CrossRefzbMATHGoogle Scholar
  12. Kyprianou, A.E.: Fluctuations of Lévy Processes with Applications, Introductory Lectures, 2nd edn. Springer, Heidelberg (2013) zbMATHGoogle Scholar
  13. Revuz, D., Yor, M.: Continuous Martingales and Brownian Motion, 3rd edn. Springer, Berlin (2004) Google Scholar
  14. Rogers, L.C.G.: A guided tour through excursions. Bull. London Math. Soc. 21, 305–341 (1989) MathSciNetCrossRefzbMATHGoogle Scholar
  15. Suprun, V.N.: Problem of ruin and resolvent of terminating processes with independent increments. Ukrainian Math. J. 28, 39–45 (1976) MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer International Publishing Switzerland 2013

Authors and Affiliations

  • Andreas E. Kyprianou
    • 1
  1. 1.Department of Mathematical SciencesUniversity of BathBathUnited Kingdom

Personalised recommendations