Advertisement

The Perturbed Sparre Andersen Model with Interest and a Threshold Dividend Strategy

  • Wei WangEmail author
Article

Abstract

In this paper, we consider a Sparre Andersen model perturbed by diffusion (in which the inter-claim times are generalized Erlang(n)-distributed) with a constant interest under a threshold dividend payment strategy. Under such a strategy, no dividends are paid if the insurer’s surplus is below a certain threshold level. When the surplus is above the threshold level, part of the premium income and all of the interest income are paid out as dividends. Integro-differential equations with certain boundary conditions for the moment generating functions and moment functions of the present value of all dividends until ruin are derived. We also derive the integro-differential equations with boundary conditions for the Gerber–Shiu functions. Explicit expressions are given in terms of some functions related to high order integro-differential equations when the inter-claim times are Erlang(2) and Erlang(1) distributed.

Keywords

Gerber–Shiu function Moment generating function Threshold dividend strategy Integro-differential equation 

AMS 2000 Subject Classifications

Primary 60K10 91B30; Secondary 60K05 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Albrecher H, Claramunt MM, Mármol M (2005a) On the distribution of dividend payments in a Sparre Andersen model with generalized Erlang(n) interclaim times. Insur Math Econ 37:324–334CrossRefzbMATHGoogle Scholar
  2. Albrecher H, Hartinger J, Tichy R (2005b) On the distribution of the dividend payment and the discounted penalty function in a risk model with a linear dividend barrier. Scand Actuar J 2:103–126CrossRefMathSciNetGoogle Scholar
  3. Avanzi B, Gerber HU, Shiu ESW (2007) Optimal dividends in the dual model. Insur Math Econ 41:111–123CrossRefzbMATHMathSciNetGoogle Scholar
  4. Cai J, Dickson DCM (2002) On the expected discounted penalty function at ruin of a surplus process with interest. Insur Math Econ 30:389–404CrossRefzbMATHMathSciNetGoogle Scholar
  5. Cai J, Yang HL (2005) Ruin in the perturbed compound Poisson risk process under interest force. Adv Appl Probab 37:819–835CrossRefzbMATHMathSciNetGoogle Scholar
  6. Cai J, Gerber HU, Yang HL (2006) Optimal dividends in an Ornstein-Uhlenbeck type model with credit and debit interest. N Am Actuar J 10(2):94–119CrossRefMathSciNetGoogle Scholar
  7. De Finetti B (1957) Su un’impostazione alternativa dell teoria collectiva del rischio. In: Transaction of the 15th international congress of actuaries, vol 2. New York, pp 433–443Google Scholar
  8. Fang Y, Wu R (2007) Optimal dividend strategy in the compound Poisson model with constant interest. Stoch Models 23:149–166CrossRefzbMATHMathSciNetGoogle Scholar
  9. Gao S, Liu ZM (2010) The perturbed compound Poisson risk model with constant interest and a threshold dividend strategy. J Comput Appl Math 233:2181–2188CrossRefzbMATHMathSciNetGoogle Scholar
  10. Gao HL, Yin CC (2008a) A perturbed risk process compounded by a geometric Brownian motion with a dividend barrier strategy. Appl Math Comput 205:454–464CrossRefzbMATHMathSciNetGoogle Scholar
  11. Gao HL, Yin CC (2008b) The perturbed Sparre Andersen model with a threshold dividend strategy. J Comput Appl Math 220:394–408CrossRefzbMATHMathSciNetGoogle Scholar
  12. Gerber HU (1981) On the probability of ruin in the presence of a linear dividend barrier. Scand Actuar J 2:105–115CrossRefMathSciNetGoogle Scholar
  13. Gerber HU, Landry B (1998) On the discounted penalty at ruin in a jump-diffusion and the perpetual put option. Insur Math Econ 22:263–276CrossRefzbMATHMathSciNetGoogle Scholar
  14. Gerber HU, Shiu ESW (2005) The time value of ruin in a Sparre Andersen model. N Am Actuar J 9(2):49–84CrossRefzbMATHMathSciNetGoogle Scholar
  15. Gerber HU, Shiu ESW (2006) On optimal dividend strategies in the compound Poisson model. N Am Actuar J 10(2):76–93CrossRefMathSciNetGoogle Scholar
  16. Li SM, Garrido J (2004) On a class of renewal risk models with a constant dividend barrier. Insur Math Econ 35:691–701CrossRefzbMATHMathSciNetGoogle Scholar
  17. Lin XS, Pavlova KP (2006) The compound Poisson risk model with a thershold dividend strategy. Insur Math Econ 38:57–80CrossRefzbMATHMathSciNetGoogle Scholar
  18. Lin XS, Willmot GE (1999) Analysis of a defective renewal eqaution arising in ruin theory. Insur Math Econ 25:63–84CrossRefzbMATHMathSciNetGoogle Scholar
  19. Lin XS, Willmot GE, Drekic S (2003) The classical risk model with a constant dividend barrier: analysis of the Gerber–Shiu discounted penalty function. Insur Math Econ 33:551–566CrossRefzbMATHMathSciNetGoogle Scholar
  20. Mitric IR, Badescu AL, Stanford DA (2012) On the absolute ruin problem in a Sparre Andersen model with constant interest. Insur Math Econ 50:167–178CrossRefzbMATHMathSciNetGoogle Scholar
  21. Paulsen J, Gjessing H (1997) Optimal choice of dividend barriers for a risk process with stochastic return on investments. Insur Math Econ 20:215–223CrossRefzbMATHMathSciNetGoogle Scholar
  22. Wan N (2007) Dividend payments with a threshold strategy in the compound Poisson risk model perturbed by diffusion. Insur Math Econ 40:509–523CrossRefzbMATHGoogle Scholar
  23. Wang GJ, Wu R (2008) The expected discounted penalty function for the perturbed compound Poisson risk process with constant interest. Insur Math Econ 42:59–64CrossRefzbMATHGoogle Scholar
  24. Yuan HL, Hu YJ (2009) The compound Poisson risk model with interest and a thershold strategy. Stoch Models 25:197–220CrossRefzbMATHMathSciNetGoogle Scholar
  25. Yuen KC, Wang GJ, Li WK (2007) The Gerber–Shiu discounted penalty function for risk process with interest and a cosntant dividend barrier. Insur Math Econ 40:104–112CrossRefzbMATHMathSciNetGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2013

Authors and Affiliations

  1. 1.College of Mathematical ScienceTianjin Normal UniversityTianjinChina

Personalised recommendations