标题： CEV模型的欧拉-马略卡逼近 显示英文标题

标题： The Euler-Maruyama approximations for the CEV model

摘要： CEV模型由随机微分方程给出 $X_t=X_0+\int_0^t\mu X_sds+\int_0^t\sigma (X^+_s)^pdW_s$，$\frac{1}{2}\le p<1$。 它具有非Lipschitz扩散系数，并且以正概率被吸收在零点。 我们展示了欧拉-马鲁亚姆方法近似值$X_t^n$在Skorokhod度量下弱收敛到过程$X_t$，$0\le t\le T$。 我们通过连续过程给出了一种新的近似方法，这使得在\cite{HZa}中弱收敛证明的一些技术条件可以被放松，该证明是基于离散时间鞅问题进行的。 我们计算了破产概率，作为这种近似的一个例子。 我们证明了通过模拟计算的破产概率不一定收敛到理论值，因为零点是极限分布的不连续点。 为了建立这种收敛性，我们使用了Levy度量，并且也通过数值计算确认了收敛性。 尽管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非Lipschitz扩散与吸收情况。 显示英文摘要

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.