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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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