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On the dual risk model with diffusion under a mixed dividend strategy. (English) Zbl 07197515
Summary: Inspired by the work of Z. Zhang and X. Han [ibid. 315, 1–12 (2017; Zbl 1427.91080)], this paper investigates a dual model with diffusion where dividends are paid under a mixed strategy. This strategy is composed of two parts: dividends will be paid continuously at a fixed rate \(\alpha > 0\) as long as the surplus process is above a fixed threshold level \(b > 0\); for a pre-specified sequence of strictly increasing periodic dividend decision times \(\{ Z_j\}_{j \geq 1}\), whenever the surplus level observed at \(Z_j\) is above \(b\), the excess value will also be paid out as dividend. In addition, ruin is declared when the observed surplus equals to 0 for the first time. The integro-differential equations satisfied by the expected present value of dividends paid up to ruin (i.e., \(V(x; b)\)) and the Laplace transform of the ruin time (i.e., \(\Phi(x; b)\)) are derived. The solutions of \(V\) and \(\Phi\) are constructed by the method of inverse Laplace transform and through some auxiliary functions. Finally, several numerical examples are provided to illustrate our results.
MSC:
60J60 Diffusion processes
60K15 Markov renewal processes, semi-Markov processes
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