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On a risk model with surplus-dependent premium and tax rates. (English) Zbl 1260.91120
Summary: In this paper, the compound Poisson risk model with surplus-dependent premium rate is analyzed in the taxation system proposed by H. Albrecher and C. Hipp [Bl. DGVFM 28, No. 1, 13–28 (2007; Zbl 1119.62103)]. In the compound Poisson risk model, Albrecher and Hipp showed that a simple relationship between the ruin probabilities in the risk model with and without tax exists. This so-called “tax identity” was later generalized to a surplus-dependent tax rate by C. Albrecher et al. [Insur. Math. Econ. 44, No. 2, 304–306 (2009; Zbl 1163.91430)]. The present paper further generalizes these results to the Gerber-Shiu function with a generalized penalty function involving the maximum surplus prior to ruin. We show that this generalized Gerber-Shiu function in the risk model with tax is closely related to the ‘original’ Gerber-Shiu function in the risk model without tax defined in a dividend barrier framework. The moments of the discounted tax payments before ruin and the optimal threshold level for the tax authority to start collecting tax payments are also examined.

MSC:
91B30 Risk theory, insurance (MSC2010)
60G55 Point processes (e.g., Poisson, Cox, Hawkes processes)
60J75 Jump processes (MSC2010)
90B05 Inventory, storage, reservoirs
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