On the time and the number of claims when the surplus drops below a certain level

Shuanming Li and Yi Lu

Scandinavian Actuarial Journal, 2016, vol. 2016, issue 5, 420-445

Abstract: In this paper, we define Tz(u)$ T_{z}(u) $ to be the first time that the surplus process drops below a certain level z$ z $ from the initial surplus u(>z)$ u({>}z) $ for a risk model with interest. A generalized Gerber–Shiu-type function is then defined based on the first time and the number of claims that the surplus drops below z$ z $ from u$ u $, and other Tz(u)$ T_{z}(u) $-related random variables. Explicit expressions for this function, when u=z$ u=z $, and when u>z$ u>z $ under exponential claims, are obtained. We then obtain the moments and probability function (with numerical examples) of the number of claims until Tz(u)$ T_{z}(u) $. We also investigate a joint transform function of the two-sided first exit time and the number of claims until then, and obtain the probability of the surplus hitting an upper level from the initial surplus without having dropped below a lower level with Erlang(2)$ (2) $ claims.

Date: 2016
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DOI: 10.1080/03461238.2014.954607

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