Moments of Discounted Dividend Payments in the Sparre Andersen Model with a Constant Dividend Barrier

ABSTRACT

We consider the Sparre Andersen risk process in the presence of a constant dividend barrier, and propose a new expected discounted penalty function which is different from that of Gerber and Shiu. We find that iteration mothed can be used to compute the values of expected discounted dividends until ruin and the new penalty function. Applying the new function and the recursion method proposed in Section 5, we obtain the arbitrary moments of discounted dividend payments until ruin.

We consider the Sparre Andersen risk process in the presence of a constant dividend barrier, and propose a new expected discounted penalty function which is different from that of Gerber and Shiu. We find that iteration mothed can be used to compute the values of expected discounted dividends until ruin and the new penalty function. Applying the new function and the recursion method proposed in Section 5, we obtain the arbitrary moments of discounted dividend payments until ruin.

KEYWORDS

Sparre Andersen Model, Expected Discounted Penalty Function, Constant Dividend Barrier, Recursion, Iteration

Sparre Andersen Model, Expected Discounted Penalty Function, Constant Dividend Barrier, Recursion, Iteration

Cite this paper

nullJ. Tan, L. Xiao, S. Liu and X. Yang, "Moments of Discounted Dividend Payments in the Sparre Andersen Model with a Constant Dividend Barrier,"*Applied Mathematics*, Vol. 2 No. 4, 2011, pp. 444-451. doi: 10.4236/am.2011.24056.

nullJ. Tan, L. Xiao, S. Liu and X. Yang, "Moments of Discounted Dividend Payments in the Sparre Andersen Model with a Constant Dividend Barrier,"

References

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[2] X. S. Lin, G. E. Willmot and S. Drekic, “The Classical Risk Models with a Constant Dividend Barrier: Analysis of the GERBER-Shiu Discounted Penalty Function,” Insurance: Mathematics and Economics, Vol. 33, No. 3, 2003, pp. 551-566.

[3] D. C. M. Dickson and H. R. Waters, “Some Optimal Dividend Problems,” Astin Bulletin, Vol. 34, No. 1, 2004, pp. 49-74. doi:10.2143/AST.34.1.504954

[4] H. U. Gerber, E. S. W. Shiu and N. Smith, “Methods for Estimating the Optimal Dividend Barrier and the Probability of Ruin,” Insurance: Mathematics and Economics, Vol. 42, No. 2, 2008, pp. 243-254. doi:10.1016/j.insmatheco.2007.02.002

[5] E. S. Anderson, “On the Collective Theory of Risk in Case of Contagion between Claims,” Bulletin of the Institute of Mathematics and Its Applications, Vol. 12, No. 2, 1957, pp. 275-279.

[6] S. Li and J. Garrido, “On a Class of Renewal Risk Models with a Constant Dividend Barrier,” Insurance: Mathe- matics and Economics, Vol. 35, No. 3, 2004, pp. 691-701. doi:10.1016/j.insmatheco.2004.08.004

[7] M. M. Claramunt, M. Marmol and R. Lacayo, “On the Probability of Reaching a Barrier in an Erlang(2) Risk Process,” Working Paper No. 24, Universitat Autonoma de Barcelona, Cerdanyola del Vallès, 2004.

[8] H. Albrecher, M. Mercgravee Claramunt and M. Marmol, “On the Distribution of Dividend Payments in a Sparre Andersen Model with Generalized Erlang(n) Interclaim Times,” Insurance: Mathematics and Economics, Vol. 37, No. 2, 2005, pp.324-334. doi:10.1016/j.insmatheco.2005.05.004

[9] Erwin Kreyszig, “Introductory Functional Analysis with Applications,” John Wiley & Sons, Hoboken, 1978, pp. 299-302.

[1] B. De Finetti, “Su un'Impostazione Alternativa Della Teoria Collettiva del Rischio,” Transactions of the XVth International Congress of Actuaries, Vol. 2, No. 1, 1957, pp. 433-443.

[2] X. S. Lin, G. E. Willmot and S. Drekic, “The Classical Risk Models with a Constant Dividend Barrier: Analysis of the GERBER-Shiu Discounted Penalty Function,” Insurance: Mathematics and Economics, Vol. 33, No. 3, 2003, pp. 551-566.

[3] D. C. M. Dickson and H. R. Waters, “Some Optimal Dividend Problems,” Astin Bulletin, Vol. 34, No. 1, 2004, pp. 49-74. doi:10.2143/AST.34.1.504954

[4] H. U. Gerber, E. S. W. Shiu and N. Smith, “Methods for Estimating the Optimal Dividend Barrier and the Probability of Ruin,” Insurance: Mathematics and Economics, Vol. 42, No. 2, 2008, pp. 243-254. doi:10.1016/j.insmatheco.2007.02.002

[5] E. S. Anderson, “On the Collective Theory of Risk in Case of Contagion between Claims,” Bulletin of the Institute of Mathematics and Its Applications, Vol. 12, No. 2, 1957, pp. 275-279.

[6] S. Li and J. Garrido, “On a Class of Renewal Risk Models with a Constant Dividend Barrier,” Insurance: Mathe- matics and Economics, Vol. 35, No. 3, 2004, pp. 691-701. doi:10.1016/j.insmatheco.2004.08.004

[7] M. M. Claramunt, M. Marmol and R. Lacayo, “On the Probability of Reaching a Barrier in an Erlang(2) Risk Process,” Working Paper No. 24, Universitat Autonoma de Barcelona, Cerdanyola del Vallès, 2004.

[8] H. Albrecher, M. Mercgravee Claramunt and M. Marmol, “On the Distribution of Dividend Payments in a Sparre Andersen Model with Generalized Erlang(n) Interclaim Times,” Insurance: Mathematics and Economics, Vol. 37, No. 2, 2005, pp.324-334. doi:10.1016/j.insmatheco.2005.05.004

[9] Erwin Kreyszig, “Introductory Functional Analysis with Applications,” John Wiley & Sons, Hoboken, 1978, pp. 299-302.