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摘要翻译：
本文提出了描述大粒子系统的随机偏微分方程的Milstein有限差分格式。通过Fourier分析，证明了无界区域上的离散在时间步长上是一阶收敛的，在空间网格尺寸上是二阶收敛的，并且离散对于边界数据是稳定的。数值实验清楚地表明，对于边值问题，同样的收敛阶也是成立的。与SPDE的标准离散化或粒子系统的直接模拟相比，以前用于SDEs的多级路径模拟显示出显著的复杂性增益。我们推导了复杂度界，并通过一个篮子信用衍生工具的应用来说明结果。
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英文标题：
《Stochastic finite differences and multilevel Monte Carlo for a class of
SPDEs in finance》
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作者：
Michael B. Giles and Christoph Reisinger
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最新提交年份：
2012
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分类信息：
一级分类：Mathematics 数学
二级分类：Numerical Analysis 数值分析
分类描述：Numerical algorithms for problems in analysis and algebra, scientific computation
分析和代数问题的数值算法，科学计算
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一级分类：Quantitative Finance 数量金融学
二级分类：Computational Finance 计算金融学
分类描述：Computational methods, including Monte Carlo, PDE, lattice and other numerical methods with applications to financial modeling
计算方法，包括蒙特卡罗，偏微分方程，格子和其他数值方法，并应用于金融建模
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英文摘要：
In this article, we propose a Milstein finite difference scheme for a stochastic partial differential equation (SPDE) describing a large particle system. We show, by means of Fourier analysis, that the discretisation on an unbounded domain is convergent of first order in the timestep and second order in the spatial grid size, and that the discretisation is stable with respect to boundary data. Numerical experiments clearly indicate that the same convergence order also holds for boundary-value problems. Multilevel path simulation, previously used for SDEs, is shown to give substantial complexity gains compared to a standard discretisation of the SPDE or direct simulation of the particle system. We derive complexity bounds and illustrate the results by an application to basket credit derivatives.
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PDF链接：
https://arxiv.org/pdf/1204.1442
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