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Limit theorems for U- and V-statistics for dependent random variables and their applications

因随机变量的 U 和 V 统计量的极限定理及其应用

基本信息

批准号：
16540124
负责人：
KANAGAWA Shuya
金额：
$ 2.46万
依托单位：
Musashi Institute of Technology
依托单位国家：
日本
项目类别：
Grant-in-Aid for Scientific Research (C)
财政年份：
2004
资助国家：
日本
起止时间：
2004 至 2007
项目状态：
已结题

项目摘要

The author investigated limit theorems for symmetric statistics(U-statistics and V-statistics) for dependent random variables using new technique by applying limit theorems for Banach space valued i.i.d. random variables. Usually well known Hoeffding's decomposition for symmetric scholastics cannot be used for symmetric statistics with non-degenerate kernels. Furthermore we tried some simulations of such dependent random variables satisfying some mixing conditions or strongly dependence. The simulation can be applied to mathematical finance under some dependent conditions.The author focused on the distribution of pseudo-random numbers which are used for numerical application of such approximate solutions and consider the error estimation of the Euler-Maruyama approximation when the distribution of underlying random variables is different from the normal distribution. Furthermore some results for stochastic differential equations with boudary conditions on multi-dimensional domains(so-called Skorohod SDE) are obtained. We define an approximate solution of stochastic differential equation(SDE) with a reflecting barrier using the penalty method and estimate error of the approximate solution. In this note we have two aims. One is to define the approximate solution using not only a sequence of increments of Brownian motion which is independent and has normal distribution but also dependent sequence that does not obey normal distribution. Another one is, to show the advantage of the penalty method, we observe sample paths of Brownian motion with a soft boundary, i.e. any path of the Brownian motion does not reflect at the bundary immediately but is absorbed for a short period according to the strength of the path getting out of the boundary.

本文应用Banach空间值独立同分布的极限定理，研究了相依随机变量对称统计量（U-统计量和V-统计量）的极限定理。随机变量通常众所周知的Hoeffding对称scholastics分解不能用于非退化核的对称统计。此外，我们还尝试了对满足某些混合条件或强相依的相依随机变量的模拟。在一定的条件下，模拟可以应用于金融数学中，本文重点讨论了用于模拟的伪随机数的分布，并考虑了当随机变量的分布不同于正态分布时，Euler-Maruyama近似的误差估计。进一步得到了多维区域上具有边界条件的随机微分方程的一些结果。利用罚函数法定义了带反射壁的随机微分方程的一个近似解，并对近似解的误差进行了估计。在本说明中，我们有两个目的。一种是用独立的正态分布布朗运动的增量序列和不服从正态分布的相依布朗运动的增量序列来定义近似解。另一个是为了显示罚函数法的优点，我们观察了具有软边界的布朗运动的样本路径，即布朗运动的任何路径都不会立即在边界处反射，而是根据离开边界的路径的强度而在短时间内被吸收。