ON THE COMPOUND POISSON RISK MODEL WITH PERIODIC CAPITAL INJECTIONS
ON THE COMPOUND POISSON RISK MODEL WITH PERIODIC CAPITAL INJECTIONS
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DOI：
10.1017/asb.2017.22
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发表时间：
2017-09
期刊：
影响因子：
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通讯作者：
Zhimin Zhang;Eric C. K. Cheung;Hailiang Yang
中科院分区：
文献类型：
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作者：
Zhimin Zhang;Eric C. K. Cheung;Hailiang Yang
摘要在保险风险模型文献(如Pafumi，1998；Dickson and Waters，2004)对资本注入策略的分析中，通常假设每当盈余变为负值时，就注入缺口，以便公司能够永远继续经营下去。最近，聂卫东等人提出了自己的观点。(2011)提出了另一种模型，即当盈余落在零和b之间时，立即注入资本，将盈余水平恢复到正水平b，而保险人仍然受到正破产概率的影响。灵感来自Albrecher等人的随机观察的想法。(2011B)，本文进一步推广了Nie等人的工作。(2011)的S模型，他假设注资只允许在一系列时间点上进行，两次注资时间服从Erlang分布(因此可以使用Asmussen等人的Erlang化技术来近似确定的时间间隔)。(2002))。当索赔金额为指数分布时，得到了Gerber-Shiu期望折现罚金函数(Gerber and Shiu，1998)和破产前注资的预期折现总成本的显式公式。导数依赖于与Erlang随机变量相关的预解密度，它被证明也允许独立感兴趣的显式表达式。我们将提供数值例子，包括在对进行资本注入的永久再保险合同定价中的应用，以及如何通过再保险最小化破产概率的演示。关于注资频率和临界水平b的预期贴现注资加上破产时适用的罚金的最小化也将用数字说明。
Abstract The analysis of capital injection strategy in the literature of insurance risk models (e.g. Pafumi, 1998; Dickson and Waters, 2004) typically assumes that whenever the surplus becomes negative, the amount of shortfall is injected so that the company can continue its business forever. Recently, Nie et al. (2011) has proposed an alternative model in which capital is immediately injected to restore the surplus level to a positive level b when the surplus falls between zero and b, and the insurer is still subject to a positive ruin probability. Inspired by the idea of randomized observations in Albrecher et al. (2011b), in this paper, we further generalize Nie et al. (2011)'s model by assuming that capital injections are only allowed at a sequence of time points with inter-capital-injection times being Erlang distributed (so that deterministic time intervals can be approximated using the Erlangization technique in Asmussen et al. (2002)). When the claim amount is distributed as a combination of exponentials, explicit formulas for the Gerber–Shiu expected discounted penalty function (Gerber and Shiu, 1998) and the expected total discounted cost of capital injections before ruin are obtained. The derivations rely on a resolvent density associated with an Erlang random variable, which is shown to admit an explicit expression that is of independent interest as well. We shall provide numerical examples, including an application in pricing a perpetual reinsurance contract that makes the capital injections and demonstration of how to minimize the ruin probability via reinsurance. Minimization of the expected discounted capital injections plus a penalty applied at ruin with respect to the frequency of injections and the critical level b will also be illustrated numerically.