Application of Advanced Integrodifferential Equations in Insurance Mathematics and Process Engineering

  • Éva Orbán-MihálykóEmail author
  • Csaba Mihálykó
Conference paper
Part of the Springer Proceedings in Mathematics & Statistics book series (PROMS, volume 94)


In this paper we consider a dual risk model with general inter-arrival time distribution and general size distribution. A special Gerber–Shiu discounted penalty function is defined and an integral equation is derived for it in the case of dependent inter-arrival times and sizes. We prove the existence and uniqueness of the solution of the integral equation in the set of bounded functions and we show that the solution tends to zero exponentially. If the density function of the inter-arrival time satisfies a linear differential equation with constant coefficients, the integral equation is transformed into an integrodifferential equation with advances in arguments and an explicit solution is given without any assumption on the size distribution. We also present a link between the Lundberg fundamental equations of the Sparre Andersen risk model and the dual risk model.


Dual risk model Advanced integrodifferential equation Process engineering LODE-type distribution 


  1. 1.
    Albrecher, H., Badescu, A., Landriualt, D.: On the dual risk model with tax payments. Insur. Math. Econ. 42, 1086–1094 (2008)CrossRefzbMATHGoogle Scholar
  2. 2.
    Albrecher, H., Constaninescu, C., Pirsic, G., Regensburger, G., Rosenkranz, M.: An algebraic operation approach to the analysis of Gerber-Shiu functions. Insur. Math. Econ. 46, 42–51 (2010)CrossRefzbMATHGoogle Scholar
  3. 3.
    Avanzi, B., Gerber, H.U., Shiu, E.S.W.: Optimal dividends in the dual model. Insur. Math. Econ. 41, 111–123 (2007)CrossRefzbMATHMathSciNetGoogle Scholar
  4. 4.
    Dong, Y.: Ruin probability for correlated negative risks sums model with Erlang process. Appl. Math. J. Chin. Univ. 24, 14–20 (2009)CrossRefzbMATHGoogle Scholar
  5. 5.
    Dong, Y., Wang, G.: Ruin probability for renewal risk model with negative risk sums. J. Ind. Manag. Optim. 2, 229 (2006)zbMATHMathSciNetGoogle Scholar
  6. 6.
    Dong, Y., Wang, G.: On a compounding assets model with positive jumps. Appl. Stoch. Model Bus. Ind. 24, 21–30 (2008)CrossRefzbMATHMathSciNetGoogle Scholar
  7. 7.
    Gerber, H.U., Shiu, E.S.W.: On the time value of ruin. N. Am. Actuar. J. 2, 48–72 (1998)CrossRefzbMATHMathSciNetGoogle Scholar
  8. 8.
    Grandell, J.: Aspects of Risk Theory. Springer, New York (1991)CrossRefzbMATHGoogle Scholar
  9. 9.
    Landriault, D., Willmot, G.: On the Gerber-Shiu discounted penalty function in the Sparre Andersen model with an arbitrary inter-claim time distribution. Insur. Math. Econ. 42, 600–608 (2008)CrossRefzbMATHMathSciNetGoogle Scholar
  10. 10.
    Li, S., Dickson, D.C.M.: The maximum surplus before ruin in an Erlang(n) risk process and related problems. Insur. Math. Econ. 38, 529–539 (2006)CrossRefzbMATHMathSciNetGoogle Scholar
  11. 11.
    Li, S., Garrido, J.: On ruin for the Erlang(n) risk process. Insur. Math. Econ. 35, 391–408 (2004)CrossRefGoogle Scholar
  12. 12.
    Li, S., Garrido, J.: On a general class of renewal risk process: analysis of the Gerber-Shiu function. Adv. Appl. Probab. 37, 836–856 (2005)CrossRefzbMATHMathSciNetGoogle Scholar
  13. 13.
    Li, S., Garrido, J.: The Gerber-Shiu function in a Sparre Andersen risk process perturbed by diffusion. Scand. Actuar. J. 3, 161–186 (2005)CrossRefMathSciNetGoogle Scholar
  14. 14.
    Ng, A.C.Y.: On a dual model with dividend threshold. Insur. Math. Econ. 44, 315–324 (2009)CrossRefzbMATHGoogle Scholar
  15. 15.
    Orbán-Mihálykó, É., Lakatos, B.G.: Intermediate storage in batch/semicontinuous processing systems under stochastic operational conditions. Comput. Chem. Eng. 28, 2493–2508 (2004)CrossRefGoogle Scholar
  16. 16.
    Orbán-Mihálykó, É., Lakatos, B.G.: On the advanced integral and differential equations of sizing procedure of storage devices. Funct. Differ. Equ. 11, 121–131 (2004)zbMATHMathSciNetGoogle Scholar
  17. 17.
    Schassberger, R.S.: Warteschlangen. Springer, Berlin (1973)CrossRefzbMATHGoogle Scholar

Copyright information

© Springer International Publishing Switzerland 2014

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of PannoniaVeszprémHungary

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