Discounted aggregate claim costs until ruin in the discrete-time renewal risk model.

*(English)*Zbl 1411.91324Summary: In this paper, we focus on analyzing the relationship between the discounted aggregate claim costs until ruin and ruin-related quantities including the time of ruin. To facilitate the evaluation of quantities of our interest as an approximation to the ones in the continuous case, discrete-time renewal risk model with certain dependent structure between interclaim times and claim amounts is considered. Furthermore, to provide explicit expressions for various moment-based joint probabilities, a fairly general class of distributions, namely the discrete Coxian distribution, is used for the interclaim times. Also, we assume a combination of geometrics claim size with arbitrary interlciam time distribution to derive a nice expression for the Gerber-Shiu type function involving the discounted aggregate claims until ruin. Consequently, the results are applied to evaluate some interesting quantities including the covariance between the discounted aggregate claim costs until ruin and the discounted claim causing ruin given that ruin occurs.

##### MSC:

91B30 | Risk theory, insurance (MSC2010) |

60K10 | Applications of renewal theory (reliability, demand theory, etc.) |

##### Keywords:

discounted aggregate claims until ruin; discrete sparre Andersen renewal risk process; discounted moment-based joint distribution; higher moments; covariance; discrete Coxian \((K_n)\) distribution; claim causing ruin
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\textit{J.-K. Woo} and \textit{H. Liu}, Methodol. Comput. Appl. Probab. 20, No. 4, 1285--1318 (2018; Zbl 1411.91324)

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