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近似是一种近似E[f(X(T)]的新格式，其中X(T)表示扩散过程，f表示具有一定规律性的函数。这个问题叫做弱近似。通过使用Kusuoka近似，我们期望可以减少在近似的最后一步产生的积分域的维数。如果我们使用拟蒙特卡罗技术，这个积分维数是一个非常关键的因素。在这个项目中，我们取得了以下成功：1。发现了一种通用算法，使我们能够轻松地将Kusuoka近似应用于sdes描述的任何扩散过程。该算法还与拟蒙特卡罗方法兼容。将该算法应用于金融衍生品定价问题，结果表明，新算法的计算速度比现有方法至少快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.