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Methodology and Computing in Applied Probability

, Volume 18, Issue 3, pp 747–764 | Cite as

Number of Jumps in Two-Sided First-Exit Problems for a Compound Poisson Process

  • Shuanming Li
  • Yi LuEmail author
  • Can Jin
Article

Abstract

In this paper, we study the joint Laplace transform and probability generating functions of two pairs of random variables: (1) the two-sided first-exit time and the number of claims by this time; (2) the two-sided smooth exit-recovery time and its associated number of claims. The joint transforms are expressed in terms of the so-called doubly-killed scale function that is defined in this paper. We also find explicit expressions for the joint density function of the two-sided first-exit time and the number of claims by this time. Numerical examples are presented for exponential claims.

Keywords

Classical risk model Two-sided first-exit time Two-sided smooth exit-recovery time Number of claims Doubly-killed scale function 

Mathematical Subject Classifications (2010)

91B30 37A50 

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References

  1. Avram F, Kyprianou AE, Pistorius MR (2004) Exit problems for spectrally negative Lévy processes and applications to (Canadized) Russian options. Ann Appl Probab 14(1):215–8C238MathSciNetCrossRefzbMATHGoogle Scholar
  2. Bertoin J (1997) Exponential decay and ergodicity of completely asymmetric Lévy processes in a finite interval. Ann Appl Probab 7(1):156–169MathSciNetCrossRefzbMATHGoogle Scholar
  3. Dickson DCM (2012) The joint distribution of the time to ruin and the number of claims until ruin in the classical risk model. Insur: Math Econ 50(3):334–337MathSciNetzbMATHGoogle Scholar
  4. Egídio dos Reis AD (1993) How long is the surplus below zero Insur: Math Econ 12:23–38MathSciNetzbMATHGoogle Scholar
  5. Gerber HU (1990) When does the surplus reach a given target Insur: Math Econ 9:115–119MathSciNetzbMATHGoogle Scholar
  6. Gerber HU, Shiu ESW (1998) On the time value of ruin. North Am Actuar J 2(1):48–78MathSciNetCrossRefzbMATHGoogle Scholar
  7. Ivanovs J (2013) A note on killing with applications in risk theory. Insur: Math Econ 52:29–34MathSciNetzbMATHGoogle Scholar
  8. Landriault D, Shi T (2014) First passage time for compound Poisson processes with diffusion: ruin theoretical and financial applications. Scand Actuar J 4:368–382MathSciNetCrossRefGoogle Scholar
  9. Li S, Lu Y (2014) The density of the time of ruin in the classical risk model with a constant dividend barrier. Ann Actuar Sci 8(1):63–78CrossRefGoogle Scholar
  10. Kadankova T, Veraverbeke N (2007) On several two-boundary problems for a particular class of Lévy processes. J Theor Probab 20:1073–1085MathSciNetCrossRefzbMATHGoogle Scholar
  11. Kuznetsov A, Kyprianou AE, Rivero V (2013) The theory of scale functions for spectrally negative Lévy processes. Lévy Matters II. Springer Lect Notes Math 2061:97–186MathSciNetCrossRefzbMATHGoogle Scholar
  12. Kyprianou AE (2013) Gerber-Shiu Risk Theory. Springer, EAA SeriesCrossRefzbMATHGoogle Scholar
  13. Perry D, Stadje W, Zacks S (2005) A two-sided first-exit problem for a compound Poisson process with a random upper boundary. Methodol Comput Appl Probab 7:51–62MathSciNetCrossRefzbMATHGoogle Scholar
  14. Shaked M, Shanthikumar J (1994) Stochastic Orders and their Applications. Academic Press Inc, Boston, MAzbMATHGoogle Scholar
  15. Stadje W, Zacks S (2003) Upper first-exit times of compound Poisson processes revisited. Probab Eng Inf Sci 17:459–465MathSciNetCrossRefzbMATHGoogle Scholar
  16. Xu Y (2012) First exit times of compound Poisson processes with parallel boundaries. Seq Anal: Des Methods Appl 31(2):135–144MathSciNetCrossRefzbMATHGoogle Scholar
  17. Zacks S (2007) Review of some functionals of compound Poisson processes and related stopping times. Methodol Comput Appl Probab 9:343–356MathSciNetCrossRefzbMATHGoogle Scholar
  18. Zacks S, Perry D, Bshouty D, Bar-Lev S (1999) Distributions of stopping times for compound Poisson processes with positive jumps and linear boundaries. Stoch Models 15:89–101MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2015

Authors and Affiliations

  1. 1.Centre for Actuarial Studies, Department of EconomicsThe University of MelbourneMelbourneAustralia
  2. 2.Department of Statistics and Actuarial ScienceSimon Fraser UniversityBurnabyCanada

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