Efficient and Practical Implementations of Cubature on Wiener Space
Efficient and Practical Implementations of Cubature on Wiener Space
复制标题
维纳空间上Cuature的高效实用实现
DOI：
--
复制
发表时间：
2011
期刊：
影响因子：
--
通讯作者：
Terry Lyons
中科院分区：
文献类型：
--
作者：
L. Gyurkó;Terry Lyons
本文探讨并实现了高阶数值格式，用于积分线性抛物型偏微分方程与逐段光滑边界数据。我们提出的高阶蒙特-卡罗方法在计算时间上给出了非常精确的近似，我们相信与精度低得多的有限差分和基本蒙特-卡罗方案相当。这些算法中的一个关键步骤似乎是将近似的阶数调整到所需的精度。使用超高次容积公式可以大大提高效率。里昂和Victoir（“Cubature on Wiener Space，Proc. R. Soc. Lond. A 460，169-198”）给出了布朗运动5次近似。我们在一维时空中将这个体积扩展到9度和11度。好处是显而易见的。
This paper explores and implements high-order numerical schemes for integrating linear parabolic partial differential equations with piece-wise smooth boundary data. The high-order Monte-Carlo methods we present give extremely accurate approximations in computation times that we believe are comparable with much less accurate finite difference and basic Monte-Carlo schemes. A key step in these algorithms seems to be that the order of the approximation is tuned to the accuracy one requires. A considerable improvement in efficiency can be attained by using ultra high-order cubature formulae. Lyons and Victoir (“Cubature on Wiener Space, Proc. R. Soc. Lond. A 460, 169–198”) give a degree 5 approximation of Brownian motion. We extend this cubature to degrees 9 and 11 in 1-dimensional space-time. The benefits are immediately apparent.