摘要翻译：
CEV模型由随机微分方程$x_t=x_0+\int_0^t\mu x_sds+\int_0^t\sigma(x^+s)^pdw_s$,$\frac{1}{2}\lep<1$给出。它具有非Lipschitz扩散系数，并以正概率在零处被吸收。在Skorokhod度量下，我们证明了Euler-Maruyama逼近$x_t^n$对过程$x_t$，$0\let\let$的弱收敛性。本文给出了一种新的连续过程逼近方法，使离散时间鞅问题弱收敛性证明中的一些技术条件得以放松。我们计算破产概率作为这种近似的一个例子。由于零点是极限分布的不连续点，所以通过仿真计算得到的破产概率不能保证收敛于理论的破产概率。为了建立这种收敛性，我们使用了Levy度量，并用数值方法证实了这种收敛性。虽然结果是针对特定模型给出的，但我们的方法适用于更一般的具有吸收的非Lipschitz扩散情况。
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英文标题：
《The Euler-Maruyama approximations for the CEV model》
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作者：
V. Abramov, F. Klebaner, R. Liptser
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最新提交年份：
2010
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分类信息：

一级分类：Mathematics        数学
二级分类：Probability        概率
分类描述：Theory and applications of probability and stochastic processes: e.g. central limit theorems, large deviations, stochastic differential equations, models from statistical mechanics, queuing theory
概率论与随机过程的理论与应用：例如中心极限定理，大偏差，随机微分方程，统计力学模型，排队论
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一级分类：Quantitative Finance        数量金融学
二级分类：Statistical Finance        统计金融
分类描述：Statistical, econometric and econophysics analyses with applications to financial markets and economic data
统计、计量经济学和经济物理学分析及其在金融市场和经济数据中的应用
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英文摘要：
  The CEV model is given by the stochastic differential equation $X_t=X_0+\int_0^t\mu X_sds+\int_0^t\sigma (X^+_s)^pdW_s$, $\frac{1}{2}\le p<1$. It features a non-Lipschitz diffusion coefficient and gets absorbed at zero with a positive probability. We show the weak convergence of Euler-Maruyama approximations $X_t^n$ to the process $X_t$, $0\le t\le T$, in the Skorokhod metric. We give a new approximation by continuous processes which allows to relax some technical conditions in the proof of weak convergence in \cite{HZa} done in terms of discrete time martingale problem. We calculate ruin probabilities as an example of such approximation. We establish that the ruin probability evaluated by simulations is not guaranteed to converge to the theoretical one, because the point zero is a discontinuity point of the limiting distribution. To establish such convergence we use the Levy metric, and also confirm the convergence numerically. Although the result is given for the specific model, our method works in a more general case of non-Lipschitz diffusion with absorbtion.
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PDF链接：
https://arxiv.org/pdf/1005.0728