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Dividends and reinsurance under a penalty for ruin. (English) Zbl 1236.91086
Summary: We find the optimal dividend strategy in a diffusion risk model under a penalty for ruin, as in [S. Thonhauser and H. Albrecher, Insur. Math. Econ. 41, No. 1, 163–184 (2007; Zbl 1119.91047)], although we allow for both a positive and a negative penalty. Furthermore, we determine the optimal proportional reinsurance strategy, when so-called expensive reinsurance is available; that is, the premium loading on reinsurance is greater than the loading on the directly written insurance. One can think of our model as taking the one in [M. I. Taksar, Math. Methods Oper. Res. 51, No. 1, 1–42 (2000; Zbl 0947.91043), Section 6] and adding a penalty for ruin.
We use the Legendre transform to obtain the optimal dividend and reinsurance strategies. Not surprisingly, the optimal dividend strategy is a barrier strategy. Also, we investigate the effect of the penalty P on the optimal strategies. In particular, we show that the optimal barrier increases with respect to P, while the optimal proportion retained and the value function decrease with respect to P. In the end, we explore the time of ruin, and find that the expected time of ruin increases with respect to P under a net profit condition.

MSC:
91B30 Risk theory, insurance (MSC2010)
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