# zbMATH — the first resource for mathematics

Randomized dividends in the compound binomial model with a general premium rate. (English) Zbl 1164.91032
In the compound binomial model, the interclaim times $$\{W_{j}\}_{j=1}^{\infty}$$ form a sequence of i.i.d. random variables with probability mass function $$f_{W}(j)=q(1-q)^{j-1}$$. The individual claim amounts $$\{X_{j}\}_{j=1}^{\infty}$$ are assumed to be a sequence of strictly positive, integer-valued and i.i.d. random variables. It is supposed that the random variables $$\{X_{j}\}_{j=1}^{\infty}$$ are distributed like a generic random variable $$X$$ with probability mass function $$f$$. The random variables $$\{X_{j}\}_{j=1}^{\infty}$$ and $$\{W_{j}\}_{j=1}^{\infty}$$ are mutually independent. The total claim amount process $$\bar S=\{S_{k}, k\in N\}$$ is defined as $$S_{k}=\sum_{j=1}^{N_{k}}X_{j}$$. The dividend-modified surplus process $$\bar U=\{U_{k}, k\in N\}$$ is defined as $$U_0=u$$ and $$U_{k}=u+ck-S_{k}-Z_{k}$$, where $$u\in N$$ is the initial surplus level, $$c\in N^{+}$$ is the level premium rate received per period and $$Z_{k}=Z_{k-1}+\sum_{i=1}^{n+1}D_{i,k}1_{\{U_{k-1}\in[b_{i-1},b_{i})\}}$$ represents the total amount of dividends paid in the first $$k$$ periods, where $$Z_0=0$$, $$D_{i,k}$$ represents the amount of dividends paid at time $$k$$ if the surplus level at time $$k-1$$ resides into the $$i$$-th surplus layer. Using the roots of generalization of Lundberg’s fundamental equation and the general theory on difference equations, the author derives an explicit expression for the Gerber-Shiu discounted penalty function with any initial surplus. An explicit expression is also obtained for the expected discounted dividend payment before ruin.

##### MSC:
 91B30 Risk theory, insurance (MSC2010)
Full Text:
##### References:
 [1] Albrecher H., North American Actuarial Journal 11 pp 43– (2007) · doi:10.1080/10920277.2007.10597447 [2] Albrecher H., Astin Bulletin (2007) [3] Badescu A., Scandinavian Actuarial Journal (2007) [4] DOI: 10.1016/S0167-6687(99)00053-0 · Zbl 1013.91063 · doi:10.1016/S0167-6687(99)00053-0 [5] Feller W., An introduction to theory and its applications 2, 2. ed. (1971) · Zbl 0219.60003 [6] DOI: 10.2143/AST.18.2.2014949 · doi:10.2143/AST.18.2.2014949 [7] Kelley W. G., Difference equations: An introduction with applications, 1. ed. (2001) · Zbl 0970.39001 [8] DOI: 10.1080/03461230510009745 · Zbl 1142.91043 · doi:10.1080/03461230510009745 [9] Lin X. S., Insurance: Mathematics and Economics (2007) [10] DOI: 10.2143/AST.19.2.2014907 · doi:10.2143/AST.19.2.2014907 [11] DOI: 10.1016/j.insmatheco.2006.01.001 · Zbl 1147.91349 · doi:10.1016/j.insmatheco.2006.01.001 [12] Willmot G. E., Lundberg approximations for compound distributions with insurance applications (2001) · Zbl 0962.62099 · doi:10.1007/978-1-4613-0111-0 [13] Zhou , X. (2005) . Classical risk model with a multi-layer premium rate (preprint) .
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.