英文标题：
《Rate of Convergence of the Probability of Ruin in the Cram\\\'er-Lundberg
  Model to its Diffusion Approximation》
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作者：
Asaf Cohen and Virginia R. Young
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最新提交年份：
2020
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英文摘要：
  We analyze the probability of ruin for the {\\it scaled} classical Cram\\\'er-Lundberg (CL) risk process and the corresponding diffusion approximation. The scaling, introduced by Iglehart \\cite{I1969} to the actuarial literature, amounts to multiplying the Poisson rate $\\la$ by $n$, dividing the claim severity by $\\sqrtn$, and adjusting the premium rate so that net premium income remains constant. %Therefore, we think of the associated diffusion approximation as being \"asymptotic for large values of $\\la$.\"   We are the first to use a comparison method to prove convergence of the probability of ruin for the scaled CL process and to derive the rate of convergence. Specifically, we prove a comparison lemma for the corresponding integro-differential equation and use this comparison lemma to prove that the probability of ruin for the scaled CL process converges to the probability of ruin for the limiting diffusion process. Moreover, we show that the rate of convergence for the ruin probability is of order $\\mO\\big(n^{-1/2}\\big)$, and we show that the convergence is {\\it uniform} with respect to the surplus. To the best of our knowledge, this is the first rate of convergence achieved for these ruin probabilities, and we show that it is the tightest one in the general case. For the case of exponentially-distributed claims, we are able to improve the approximation arising from the diffusion, attaining a uniform $\\mO\\big(n^{-k/2}\\big)$ rate of convergence for arbitrary $k \\in \\N$. We also include two examples that illustrate our results.
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中文摘要：
我们分析了{\\it scaled}经典Cram\\er-Lundberg（CL）风险过程的破产概率和相应的扩散近似。Iglehart{I1969}在精算文献中引入的比例相当于将泊松率$\\la$乘以n$，将索赔严重性除以$\\sqrtn$，并调整保费率，使净保费收入保持不变。%因此，我们认为相关的扩散近似是“对于$\\la$的大值是渐近的”我们是第一个使用比较方法证明规模CL过程破产概率的收敛性并推导收敛速度的人。具体来说，我们证明了相应积分微分方程的一个比较引理，并利用这个比较引理证明了标度CL过程的破产概率收敛于极限扩散过程的破产概率。此外，我们还证明了破产概率的收敛速度为$\\mO\\big（n ^{-1/2}\\big）$，并且我们证明了关于盈余的收敛是{\\it一致的}。据我们所知，这是这些破产概率的第一个收敛速度，我们证明了它在一般情况下是最紧的。对于指数分布索赔的情况，我们能够改进由扩散产生的近似值，对于任意的$k，我们可以获得一致的$mO \\ big（n ^{-k/2}\\ big）$收敛速度。我们还包括两个例子来说明我们的结果。
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分类信息：

一级分类：Mathematics        数学
二级分类：Optimization and Control        优化与控制
分类描述：Operations research, linear programming, control theory, systems theory, optimal control, game theory
运筹学，线性规划，控制论，系统论，最优控制，博弈论
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一级分类：Quantitative Finance        数量金融学
二级分类：Risk Management        风险管理
分类描述：Measurement and management of financial risks in trading, banking, insurance, corporate and other applications
衡量和管理贸易、银行、保险、企业和其他应用中的金融风险
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