Additive functional of one-dimensional diffusion processes

一维扩散过程的加性泛函

基本信息

批准号：
17540105
负责人：
KASAHARA Yuji
金额：
$ 2.22万
依托单位：
University of Tsukuba
依托单位国家：
日本
项目类别：
Grant-in-Aid for Scientific Research (C)
财政年份：
2005
资助国家：
日本
起止时间：
2005 至 2007
项目状态：
已结题

项目摘要

We studied mainly the long-time asymptotic behavior of additive functionals, especially the occupation times on the positie half line, of one-dimensional diffusion processes. Historically, this problem is well known for Brownian motions and random walks and the limiting distribution obeys the are-sine law. This result has been extended in various ways by many authors. Among them J. Lamperti found the all possible limiting distributions for stochasitic processes with discrete time parameter and he also succeeded to determine the domain of attraction. Although his theorem does not include the case of one-dimensional diffusions, a similar results is shown by S. Watanabe. Many probabilists are still interested in these classical results in connection with financial theory. In our research we studied similar problems for one-dimensional diffusion processes and random walks with random drifts (I. e., in random environments). Our main results are the following: (1) A certain kind of Zero-one law holds. That is, under some technical conditions, the time spent on the positive side converges in distribution to a Bernoulli random variable almost surely. (2) In that case, if the environment is of the stable-type, the time spent on the positive side converges in law to a certain non-degenerate distribution. These results were obtained with S. Watanabe and will be published in Stochastic Processes and its Applications. Another significant result is the following. Y. Yano, et.al. recently proved a functional limit theorem for Lamperti's classical theorem for the occupation times of the positive side. However, they excluded the extreme case of index zero. Our result is that, in such a case, we obtain a functional limit theorem under a non-linear normalization. This result is a joint work with S. Suzuki and published in Proc. Of Japan Acad.

我们主要研究一维扩散过程的加性泛函的长时渐近行为，特别是在正半线上的占据时间。从历史上看，这个问题因布朗运动和随机游走而众所周知，并且极限分布服从正弦定律。许多作者以各种方式扩展了这一结果。其中J. Lamperti发现了具有离散时间参数的随机过程的所有可能的极限分布，并且成功地确定了吸引力域。尽管他的定理不包括一维扩散的情况，但 S. Watanabe 也给出了类似的结果。许多概率学家仍然对这些与金融理论相关的经典结果感兴趣。在我们的研究中，我们研究了一维扩散过程和具有随机漂移的随机游走（即在随机环境中）的类似问题。我们的主要结果如下：（1）某种零一定律成立。也就是说，在某些技术条件下，花费在正侧的时间几乎肯定会收敛于伯努利随机变量的分布。 (2) 在这种情况下，如果环境是稳定型的，则在正侧花费的时间按规律收敛到某个非简并分布。这些结果是与 S. Watanabe 一起获得的，并将发表在《随机过程及其应用》中。另一个重要结果如下。 Y.Yano 等人。最近证明了 Lamperti 经典定理的正方占据时间的泛函极限定理。然而，他们排除了索引为零的极端情况。我们的结果是，在这种情况下，我们得到了非线性归一化下的泛函极限定理。该结果是与 S. Suzuki 的合作成果，并发表在 Proc 上。日本科学院.