# American Institute of Mathematical Sciences

July  2020, 16(4): 1967-1986. doi: 10.3934/jimo.2019038

## On a perturbed compound Poisson risk model under a periodic threshold-type dividend strategy

 1 School of Economics, Southwest University of Political Science and Law, Chongqing 401120, China 2 College of Mathematics and Statistics, Chongqing University, Chongqing 401331, China

* Corresponding author: Zhimin Zhang

Received  July 2018 Revised  November 2018 Published  May 2019

Fund Project: The research of Xuanhua Peng was supported by the Chongqing Social Science Planning Project (No. 2017YBGL151), the Chongqing Municipal Education Commission Humanities and Social Sciences Research Project (No. 18SKGH006) and the Southwest University of Political Science and Law Research Project (No. 2018XZQN-35). The research of Zhimin Zhang was supported by the National Natural Science Foundation of China (Nos. 11471058, 11871121), MOE (Ministry of Education in China) Project of Humanities and Social Sciences (No. 16YJC910005) and Fundamental Research Funds for the Central Universities (No. 2018CDQYST0016)

In this paper, we model the insurance company's surplus flow by a perturbed compound Poisson model. Suppose that at a sequence of random time points, the insurance company observes the surplus to decide dividend payments. If the observed surplus level is larger than the maximum of a threshold $b>0$ and the last observed level (after dividends payment if possible), then a fraction $0<\theta<1$ of the excess amount is paid out as a lump sum dividend. We assume that the solvency is also discretely monitored at these observation times, so that the surplus process stops when the observed value becomes negative. Integro-differential equations for the expected discounted dividend payments before ruin and the Gerber-Shiu expected discounted penalty function are derived, and solutions are also analyzed by Laplace transform method. Numerical examples are given to illustrate the applicability of our results.

Citation: Xuanhua Peng, Wen Su, Zhimin Zhang. On a perturbed compound Poisson risk model under a periodic threshold-type dividend strategy. Journal of Industrial & Management Optimization, 2020, 16 (4) : 1967-1986. doi: 10.3934/jimo.2019038
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##### References:
$V(u;b)$ as a function of $u$: (a) Exponential; (b) Erlang(2); (c) Mixture of exponentials; (d) Compound of exponentials
$V(u;b)$ as a function of $b$: (a) Exponential; (b) Erlang(2); (c) Mixture of exponentials; (d) Compound of exponentials
$\phi(u;b)$ as a function of $u$: (a) Exponential; (b) Erlang(2); (c) Mixture of exponentials; (d) Combination of exponentials
$\phi(u;b)$ as a function of $b$: (a) Exponential; (b) Erlang(2); (c) Mixture of exponentials; (d) Combination of exponentials
The Gerber-Shiu function $\phi(u;b)$: (a) $b = 5$; (b) $u = 5$
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