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


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.


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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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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