Dividend moments in the dual risk model exact and approximate approaches.

*(English)*Zbl 1256.91026Summary: In the classical compound Poisson risk model, it is assumed that a company (typically an insurance company) receives premium at a constant rate and pays incurred claims until ruin occurs. In contrast, for certain companies (typically those focusing on invention), it might be more appropriate to assume expenses are paid at a fixed rate and occasional random income is earned. In such cases, the surplus process of the company can be modelled as a dual of the classical compound Poisson model, as described in [B. Avanzi et al., Insur. Math. Econ. 41, No. 1, 111–123 (2007; Zbl 1131.91026)]. Assuming further that a barrier strategy is applied to such a model (i.e., any overshoot beyond a fixed level caused by an upward jump is paid out as a dividend until ruin occurs), we are able to derive integro-differential equations for the moments of the total discounted dividends as well as the Laplace transform of the time of ruin. These integro-differential equations can be solved explicitly assuming the jump size distribution has a rational Laplace transform. We also propose a discrete-time analogue of the continuous-time dual model and show that the corresponding quantities can be solved for explicitly leaving the discrete jump size distribution arbitrary. While the discrete-time model can be considered as a stand-alone model, it can also serve as an approximation to the continuous-time model. Finally, we consider a generalization of the so-called Dickson-Waters modification in optimal dividends problems by maximizing the difference between the expected value of discounted dividends and the present value of a fixed penalty applied at the time of ruin.

##### MSC:

91B30 | Risk theory, insurance (MSC2010) |

91G80 | Financial applications of other theories |

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\textit{E. C. K. Cheung} and \textit{S. Drekic}, ASTIN Bull. 38, No. 2, 399--422 (2008; Zbl 1256.91026)

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##### References:

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