Construction of new algorithms for numerical weak approximation of Diffusion Processes by Kusuoka scheme and their applications to Finance problems

(1)我们已经成功地找到并构造了新的扩散过程的高阶数值近似算法。该算法基于Kusuoka近似理论。它们具有数值鲁棒性和通用性，即几乎可以应用于所有扩散。(2)我们将算法应用于一些金融问题，并在以下方面取得了显著的改进：(a)非常快的计算（至少比最先进的方法快100倍）(b)鲁棒离散化(c)通用性。我们已经开发了算法的计算机程序。

(1) We have succeeded in finding and constructing new higher-order numerical approximation algorithms for diffusion processes. The algorithms are based on the theory of Kusuoka approximation. They enjoy both the numerical robustness and the universality, that is, we can apply them for almost all diffusions. (2) We applied the algorithms to some finance problems and achieved remarkable improvements in : (a) very fast calculation (at least 100 times faster than the state of the art methods) (b) robust discretization (c) universality. (3) We have developed the computer program of the algorithms.