Gerber-Shiu function of a discrete risk model with and without a constant dividend barrier

Abstract

We consider the discrete risk model with exponential claim sizes. We derive the finite explicit elementary expression for the joint density function of three characteristics: the time of ruin, the surplus immediately before ruin, and the deficit at ruin. By using the explicit joint density function, we give a concise expression for the Gerber-Shiu function with no dividends. Finally, we obtain an integral equation for the Gerber-Shiu function under the barrier dividend strategy. The solution can be expressed as a combination of the Gerber-Shiu function without dividends and the solution of the corresponding homogeneous integral equation. This latter function is given clearly by means of the Gerber-Shiu function without dividends.

Keywords

Discrete risk model / Gerber-Shiu function / time of ruin / surplus before ruin / deficit at ruin / dividend

References

[1] Chan W S, Zhang L Z. Direct derivation of finite-time ruin probabilities in the discrete risk model with exponential or geometric claims. N Am Actuar J, 2006, 10(4): 269-279

[2] Cheng S, Gerber H U, Shiu E S W. Discounted probabilities and ruin theory in the compound binomial model. Insurance Math Econom, 2000, 26(2): 239-250

[3] Claramunt M M, Marmol M, Alegre A. A note on the expected present value of dividends with a constant barrier in the discrete time model. Bull Swiss Assoc Actuaries, 2003, 2: 149-159

[4] De Finetti B. Su un’impostazione alternative della teoria collettiva del rischio. Transactions of the XVth International congress of actuaries, 1957, 2: 433-443

[5] Dickson D C M. Insurance Risk and Ruin. New York: Cambridge University Press, 2004

[6] Dickson D C M, Water H R. Some optimal dividend problems. Astin Bull, 2004, 34(1): 49-74

[7] Gerber H U, Shiu E S W. On the time value of ruin. N Am Actuar J, 1998, 2(1): 48-78

[8] Gerber H U, Shiu E S W, Smith N. Methods for estimating the optimal dividend barrier and the probability of ruin. Insurance Math Econom, 2008, 42: 243-254

[9] Li S M. Distributions of the surplus before ruin, the deficit at ruin and the claim causing ruin in a class of discrete time risk model. Scand Actuar J, 2005, 105: 241-260

[10] Li S M. On a class of discrete time renewal risk model. Scand Actuar J, 2005, 105: 271-284

[11] Li S M, Lu Y, Garrido J. A review of discrete-time risk models. Rev R Acad Cien Esrie A Mat, 2009, 103(2): 321-337

[12] Lin X S, Willmot G E, Drekic S. The classical risk model with a constant dividend barrier: Analysis of the Gerber-Shiu discounted penalty function. Insurance Math Econom, 2003, 33: 551-566

[13] Pavlova K P, Willmot G E. The discrete stationary renewal risk model and the Gerber-Shiu discounted penalty function. Insurance Math Econom, 2004, 35: 267-277

[14] Tan J Y, Yang X Q. The compound binomial model with a constant dividend barrier and periodically paid dividends. J Syst Sci Complex, 2012, 5: 67-177

RIGHTS & PERMISSIONS

Higher Education Press and Springer-Verlag Berlin Heidelberg