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Gerber–Shiu analysis of a risk model with capital injections

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Abstract

We consider the risk model with capital injections studied by Nie et al. (Ann Actuar Sci 5:195–209, 2011; Scand Actuar J 2015:301–318, 2015). We construct a Gerber–Shiu function and show that whilst this tool is not efficient for finding the ultimate ruin probability, it provides an effective way of studying ruin related quantities in finite time. In particular, we find a general expression for the joint distribution of the time of ruin and the number of claims until ruin, and find an extension of Prabhu’s (Ann Math Stat 32:757–764, 1961) formula for the finite time survival probability in the classical risk model. We illustrate our results in the case of exponentially distributed claims and obtain some interesting identities. In particular, we generalise results from the classical risk model and prove the identity of two known formulae for that model.

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References

  1. Abramowitz M, Stegun IA (1965) Handbook of mathematical functions. Dover, New York

    MATH  Google Scholar 

  2. Albrecher H, Boxma O (2005) On the discounted penalty function in a Markov-dependent risk model. Insur Math Econ 37:650–672

    Article  MathSciNet  MATH  Google Scholar 

  3. Dickson DCM (1998) On a class of renewal risk processes. North Am Actuar J 2(3):60–73

    Article  MathSciNet  MATH  Google Scholar 

  4. Dickson DCM (2005) Insurance risk and ruin. Cambridge University Press, Cambridge

    Book  MATH  Google Scholar 

  5. Dickson DCM (2007) Some finite time ruin problems. Ann Actuar Sci 2:217–232

    Article  Google Scholar 

  6. 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 50:334–337

    Article  MathSciNet  MATH  Google Scholar 

  7. Dickson DCM, Hipp C (2001) On the time to ruin for Erlang(2) risk processes. Insur Math Econ 29:333–344

    Article  MathSciNet  MATH  Google Scholar 

  8. Dickson DCM, Hughes BD, Lianzeng Z (2005) The density of the time to ruin for a Sparre Andersen process with Erlang arrivals and exponential claims. Scand Actuar J 2005:358–376

    Article  MathSciNet  MATH  Google Scholar 

  9. Dickson DCM, Li S (2010) Finite time ruin problems for the Erlang(2) risk model. Insur Math Econ 46:12–18

    Article  MathSciNet  MATH  Google Scholar 

  10. Drekic S, Willmot GE (2003) On the density and moments of the time to ruin with exponential claims. ASTIN Bull 33:11–21

    Article  MathSciNet  MATH  Google Scholar 

  11. Erdélyi A (ed) (1954) Tables of integral transforms, vol 1. McGraw-Hill, New York

    MATH  Google Scholar 

  12. Gerber HU, Goovaerts MJ, Kaas R (1987) On the probability and severity of ruin. ASTIN Bull 17:151–163

    Article  Google Scholar 

  13. Gerber HU, Shiu ESW (1998) On the time value of ruin. North Am Actuar J 2(1):48–78

    Article  MathSciNet  MATH  Google Scholar 

  14. Gerber HU, Shiu ESW (2005) The time value of ruin in a Sparre Andersen model. North Am Actuar J 9(2):1–21

    Article  MathSciNet  MATH  Google Scholar 

  15. Gradshteyn IS, Ryzhnik IM (2007) Table of integrals, series, and products, 7th edn. Academic Press, San Diego

    Google Scholar 

  16. Graham RL, Knuth DE, Patashnik O (1994) Concrete mathematics, 2nd edn. Addison-Wesley, Upper Saddle River

    MATH  Google Scholar 

  17. Landriault D, Shi T, Willmot GE (2011) Joint density involving the time to ruin in the Sparre Andersen risk model under exponential assumptions. Insur Math Econ 49:371–379

    Article  MathSciNet  MATH  Google Scholar 

  18. Li S, Lu Y (2008) The decompositions of the discounted penalty functions and dividends-penalty identity in a Markov-modulated risk model. ASTIN Bull 38:53–71

    Article  MathSciNet  MATH  Google Scholar 

  19. Lin XS, Willmot GE, Drekic S (2003) The classical Poisson risk model with a constant dividend barrier: analysis of the Gerber–Shiu discounted penalty function. Insur Math Econ 33:551–566

    Article  MathSciNet  MATH  Google Scholar 

  20. Nie C, Dickson DCM, Li S (2011) Minimizing the ruin probability through capital injections. Ann Actuar Sci 5:195–209

    Article  Google Scholar 

  21. Nie C, Dickson DCM, Li S (2015) The finite time ruin probability in a risk model with capital injections. Scand Actuar J 2015:301–318

    Article  MathSciNet  Google Scholar 

  22. Panjer HH (1981) Recursive evaluation of a family of compound distributions. ASTIN Bull 12:22–26

    Article  MathSciNet  Google Scholar 

  23. Panjer HH, Willmot GE (1992) Insurance risk models. Society of Actuaries, Schaumburg

    Google Scholar 

  24. Prabhu NU (1961) On the ruin problem of collective risk theory. Ann Math Stat 32:757–764

    Article  MathSciNet  MATH  Google Scholar 

  25. Seal HL (1978) Survival probabilities—the goal of risk theory. Wiley, New York

    MATH  Google Scholar 

  26. Willmot GE (2015) On a partial differential equation of Seal’s type. Insur Math Econ 62:54–61

    Article  MathSciNet  MATH  Google Scholar 

Download references

Acknowledgments

The authors are grateful to the referees for constructive comments which improved the quality of this paper; in particular, the Proof of Theorem 3.1 was greatly improved by a referee’s comments.

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Correspondence to David C. M. Dickson.

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Dickson, D.C.M., Qazvini, M. Gerber–Shiu analysis of a risk model with capital injections. Eur. Actuar. J. 6, 409–440 (2016). https://doi.org/10.1007/s13385-016-0131-1

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  • DOI: https://doi.org/10.1007/s13385-016-0131-1

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