英语翻译Abstract.Traditional finite-time convergence theory for

英语翻译
Abstract.Traditional finite-time convergence theory for numerical methods applied to stochastic differential equations(SDES) requires a global Lipschitz assumption on the drift and diffusion coefficients.In practice,many important SDE models satisfy only a local Lipschitz property and,since Brownian paths can make arbitrarily large excursions,the global Lipschitz-based theory is not directly relevant.In this work we prove strong convergence results under less restrictive conditions.First we give a convergence result for Euler-Maruyama requiring only that the SDE is locally Lipschitz and that the pth moments of the exact and numerical solution are bounded for some p>2.As an application of this general theory we show that an implicit variant of Euler-Maruyama converges if the diffusion coefficient is globally Lipschitz,but the drift coefficient satisfies only a one-sided Lipschitz condition; this is achieved by showing that the implicit method has bounded moments and may be viewed as an Euler-Maruyama approximation to a perturbed SDE of the same form.Second,we show that the optimal rate of convergence can be recovered if the drift coefficient is also assumed to behave like a polynomial.

Abstract.
抽象的.
Traditional finite-time convergence theory for numerical methods applied to stochastic differential equations(SDES) requires a global Lipschitz assumption on the drift and diffusion coefficients.
传统的限定时间的辐合理论为数值方法应用于随机微分方程SDES)需要一个全球Lipschitz假设的漂移和扩散系数.
In practice,many important SDE models satisfy only a local Lipschitz property and,since Brownian paths can make arbitrarily large excursions,the global Lipschitz-based theory is not directly relevant.
在实践中,许多重要的钻Lipschitz模型满足只有当地资产,并自布朗路径可以使任意大的远足、全球Lipschitz-based理论不直接相关.
In this work we prove strong convergence results under less restrictive conditions.
在这部作品中我们证明强收敛结果下少一些限制条件.
First we give a convergence result for Euler-Maruyama requiring only that the SDE is locally Lipschitz and that the pth moments of the exact and numerical solution are bounded for some p>2.
首先我们给一个收敛效果,为Euler-Maruyama要求只,局部地Lipschitz钻,镀通孔失效的时刻准确、数值解有界对于一些p > 2.
As an application of this general theory we show that an implicit variant of Euler-Maruyama converges if the diffusion coefficient is globally Lipschitz,but the drift coefficient satisfies only a one-sided Lipschitz condition; this is achieved by showing that the implicit method has bounded moments and may be viewed as an Euler-Maruyama approximation to a perturbed SDE of the same form.
作为一个应用程序的一般理论,我们显示一个隐含的变种Euler-Maruyama收敛的扩散系数,但全球Lipschitz漂移系数满足只片面Lipschitz状况;达到这个目标,显示了隐式方法的时候,也有可能有界,被视作一种Euler-Maruyama逼近的摄动钻形像、都是一样.
Second,we show that the optimal rate of convergence can be recovered if the drift coefficient is also assumed to behave like a polynomial.
第二,我们发现的最优率收敛可恢复的漂移系数是如果也假定的行为像一个多项式.