Stochastic fuzzy analysis and its applications

随机模糊分析及其应用

• 批准号: 09440085
• 负责人: OGURA Yukio
• 金额: $ 5.31万
• 依托单位: Saga University
• 依托单位国家: 日本
• 项目类别: Grant-in-Aid for Scientific Research (B)
• 财政年份: 1997
• 资助国家: 日本
• 起止时间: 1997 至 1999
• 项目状态: 已结题
• 来源: https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-09440085/
• 关键词: fuzzy analysis; fuzzy martingale; random set; random fuzzy set; graph onvergence; separability for convergence; law of large numbers; optional stopping theorem; デファジィフィケイション; ファジィマルチシゲール; ファジィマルチンゲ-ル; Kudo-Aumann積分; 優マルチンゲ-ル; 劣マルチンゲ-ル

Former study on limit theorems for random fuzzy sets was mainly for the case of compact level sets. This is due to its method which is based on the embedding theorem of convex compact sets into the space of bounded continuous functions on the unit closed ball in the dual space. The object of this project is to obtain limit theorems for wider class of random fuzzy sets, extending the former study.There are two cases where the compactness condition fails. One is the case of bounded but non compact sets, which occurs in infinite dimension cases. The other is that of unbounded sets, which occurs even in finite dimensional cases. In the former case, the first problem one may encounter is the closedness of the Aumann integrals. This integral is fundamental in the study of random sets or random fuzzy sets, and although a counter example for the closeness was known, a good sufficient condition was not known. In this project, we obtained that the Aumann integral is closed if the underlying spac … More e is a reflexive Banach space. We next dealt with the concept of convergence, which is significant in both former and latter cases. In most of former works, the fuzzy sets were treated as simple collections of level sets and the topologies sometimes differs from intuition. In this project, we introduced a new concept called graph convergence which fits intuition more, and improved various convergence theorems. Finally, we also obtained separability theorem for graph convergence, which ensures that one have only to check the convergence of level sets for countably many levels for a proof of convergence of random fuzzy sets.Based on the former theoretical advance, we obtained or improved various limit theorems on random sets and/or random fuzzy sets, which covers law of large numbers, limit theorems for martingale and s-martingales, regular representation theorem for martingales, optional sampling theorems, approximation of random fuzzy sets and representation theorem of Gaussian random fuzzy sets. Less

以往对随机模糊集极限定理的研究主要针对紧集水平集的情形。这是由于它的方法是基于凸紧集到对偶空间中单位闭球上的有界连续函数空间的嵌入定理。这个项目的目的是得到更广泛的一类随机模糊集的极限定理，扩展了以前的研究。紧性条件在两种情况下失效。一种是有界但非紧集的情况，它发生在无限维的情况下。另一种是无界集的情形，这种情形即使在有限维的情况下也会发生。在前一种情况下，人们可能遇到的第一个问题是Aumann积分的闭性。这个积分在随机集或随机模糊集的研究中是基本的，虽然已经知道了贴近度的反例，但还不知道一个好的充分条件。在这个项目中，我们得到了奥曼积分是闭的，如果基础空间…更多的e是自反Banach空间。我们接下来讨论了收敛的概念，这在前一种情况和后一种情况下都很重要。在以往的工作中，模糊集大多被视为水平集的简单集合，其拓扑有时与直觉不同。在这个项目中，我们引入了一个更符合直觉的图收敛的新概念，并改进了各种收敛定理。在此基础上，我们得到或改进了关于随机集和/或随机模糊集的各种极限定理，包括大数定律、鞅和S-鞅的极限定理、鞅的正则表示定理、可选抽样定理、随机模糊集的逼近定理和高斯随机模糊集的表示定理。较少

项目成果

H.Sugita: "Limit theorem for symmetric statistics with respect to Weyl transformation : Disap-pearance of dependency" Jour.Math.Kyoto Univ.38. (1998)

H.Sugita：“关于 Weyl 变换的对称统计的极限定理：依赖性消失”Jour.Math.Kyoto Univ.38。

S.Li: "An optional sampling theorem for fuzzy valued martingales" IFSA'97 Prague Proceedings. IV. 9-14 (1997)

S.Li：“模糊值鞅的可选抽样定理”IFSA97 布拉格会议记录。

I.Mitoma: "One loop approximation of the Chern-Simons integral"in "The Volume in Honor of 70th Birthday of T.Hida". (to appear).

I.Mitoma：“陈-西蒙斯积分的一个循环近似”，载于“纪念 T.Hida 70 岁生日卷”。

S.Li: "Convergence of set valued sub-and super-martingales and smartingales in the Kuratowski-Mosco sense" Ann.Prob.26. 1384-1402 (1998)

S.Li：“Kuratowski-Mosco 意义上的设定值子鞅和超鞅以及鞅的收敛”Ann.Prob.26。

S.Li: "Convergence of set valued and fuzzy valued martingales in the Kuratowski-Mosco sense" in"Proceedings Seventh International Conference IPMU". 72-29 (1998)

S.Li：《Kuratowski-Mosco 意义上的集值鞅和模糊值鞅的收敛》，《第七届 IPMU 国际会议论文集》。