A new weak approximation scheme of diffusion and its application to Finance

这个项目的目的是建立Kusuoka近似。Kusuoka逼近是一种新的逼近E[f（X（T））]的格式，其中X（t）表示扩散过程，f是具有某种正则性的函数。这个问题称为弱近似。通过使用Kusuoka近似，我们可以减少在近似的最后一步中出现的积分域的维数。如果我们使用准蒙特卡罗技术，这个积分维数是一个非常关键的因素。在本项目中，我们取得了以下成果：1.发现了一个通用的算法，使我们能够很容易地将Kusuoka近似应用于任何由SDEs描述的扩散过程。上述算法与拟蒙特卡罗方法也是兼容的.将该算法应用于金融衍生品定价问题，结果表明，该算法比现有方法至少快800倍。

The object of this project is to establish the Kusuoka approximation. Kusuoka approximation is a new scheme that approximate E[f(X(T)] where X(t) denotes a diffusion process and f a function with some regularity. This problem is called weak approximation. By using the Kusuoka approximation, it is expected that we can reduce the number of the dimension of the domain of integration which arises in the last step of the approximation. This integral dimension is a very critical factor if we use quasi-Monte Carlo techniques. In this project, we have achieved the following successes :1. The discovery of a versatile algorithm that enables us to apply the Kusuoka approximation easily to any diffusion processes described by SDEs.2. The algorithm above also is compatible with quasi-Monte Carlo method.3. We have applied the algorithm to financial derivative pricing problem and showed that our new algorithm makes at least 800 times faster calculation than existing methods.