非紧模糊数空间上的模糊随机变量
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摘要
本文研究取非紧支集模糊数值的模糊随机变量，所谓非紧，即把传统模糊数要求的紧支撑条件去掉。作为理论基础，本文首先在非紧模糊数集上引入广义距离并证明了其完备性与可分性；其次，本文比较了几种非紧模糊随机变量的定义，其结果表明文中提到的几种非紧模糊随机变量就可测性而言是相同的；再次，讨论了非紧模糊随机变量的方差、协方差及其性质；最后初步涉及非紧模糊随机过程的理论。
第一章为模糊随机变量内容概述。内容包括模糊随机变量应用背景的介绍，模糊随机变量理论与应用发展的介绍和本文所研究问题的介绍。
第二章是关于非紧模糊数空间的内容，介绍了非紧模糊数的基本概念和非紧模糊数的广义距离空间。与可分不完备具紧支集模糊数的距离空间相比较，这里定义的广义距离空间是完备可分的。
第三章是本文的核心，研究了取非紧支集模糊数值的模糊随机变量，其内容为：首先就模糊随机变量的几种定义进行了比较，这里的结论推广了文《A unified approach to fuzzy random variables》(Volker Kr(?)tschmer,Fuzzy Sets and Systems，2001，123，1-9)的部分结果，其次介绍非紧p-阶模糊随机变量的有关内容。不难看出，非紧p-阶模糊随机变量的定义是对p-阶R~n值随机变量概念的延拓。然后用一种新方法对一类非紧模糊随机变量定义了期望。
第四章涉及非紧二阶模糊随机过程及其应用。这里主要就工程上易于应用的非紧二阶模糊随机变量和非紧二阶模糊随机过程及其一些初步应用作了介绍。文中的结果推广了《The variance and covariance of fuzzy random variables and their applications》(Feng Yuhu. et al, Fuzzy Sets and Systems, 2001, 120, 487-497.)的部分结论。最后，我们讨论了右闭右连续模糊鞅选择表示。
The paper mainly deals with fuzzy random variables that take values on noncompact fuzzy number space. Firstly, a general metric is defined on noncompact fuzzy number space and its completeness and separability are proved. Secondly, it turns out that some different concepts of noncompact fuzzy random variables are unified as the measurability is concerned. Furthermore, the variance and covariance of noncompact fuzzy random variables are introduced, and their properties are discussed. Finally, the fuzzy stochastic processes are also touched upon.
The paper is organized as follows:
Chapter 1 presents a summary of fuzzy random variables. Topics include a brief introduction of the idea of applying fuzzy random variables, an introduction of literature on fuzzy random variables and the problems considered in this thesis.
Noncompact fuzzy number space is introduced in Chapter 2. A generalized metric is defined on the noncompact fuzzy number set. Some properties and theorems about noncompact fuzzy numbers are discussed. Compared with the separability and incompleteness of compact fuzzy number space, the generalized metric defined here is proved to be complete and separable.
Noncompact fuzzy random variables are introduced in Chapter 3. We have compared some different concepts of noncompact fuzzy random variables. As the result, it will be shown that these concepts are unified as the measurability is concerned. The study extends some results of "A unified approach to fuzzy random variables (Volker Kratschmer, Fuzzy Sets and
Systems, 2001, 123, 1-9)". The concepts and properties of noncompact p-order fuzzy random variables are presented as extension of the p-order random vector. Finally, we defined the expectation of noncompact fuzzy random variables by a new method.
Noncompact fuzzy stochastic processes and their applications are put in Chapter 4. Some properties of noncompact 2-order fuzzy random variables and their applications are discussed. The results extend the study of "The variance and covariance of fuzzy random variables and their applications
(Feng Yuhu.et al, Fuzzy Sets and Systems, 2001, 120, 487-497.) ". In addition, we get a result about the right-closed-right-continuous fuzzy martingale.
第一章为模糊随机变量内容概述。内容包括模糊随机变量应用背景的介绍，模糊随机变量理论与应用发展的介绍和本文所研究问题的介绍。
第二章是关于非紧模糊数空间的内容，介绍了非紧模糊数的基本概念和非紧模糊数的广义距离空间。与可分不完备具紧支集模糊数的距离空间相比较，这里定义的广义距离空间是完备可分的。
第三章是本文的核心，研究了取非紧支集模糊数值的模糊随机变量，其内容为：首先就模糊随机变量的几种定义进行了比较，这里的结论推广了文《A unified approach to fuzzy random variables》(Volker Kr(?)tschmer,Fuzzy Sets and Systems，2001，123，1-9)的部分结果，其次介绍非紧p-阶模糊随机变量的有关内容。不难看出，非紧p-阶模糊随机变量的定义是对p-阶R~n值随机变量概念的延拓。然后用一种新方法对一类非紧模糊随机变量定义了期望。
第四章涉及非紧二阶模糊随机过程及其应用。这里主要就工程上易于应用的非紧二阶模糊随机变量和非紧二阶模糊随机过程及其一些初步应用作了介绍。文中的结果推广了《The variance and covariance of fuzzy random variables and their applications》(Feng Yuhu. et al, Fuzzy Sets and Systems, 2001, 120, 487-497.)的部分结论。最后，我们讨论了右闭右连续模糊鞅选择表示。
The paper mainly deals with fuzzy random variables that take values on noncompact fuzzy number space. Firstly, a general metric is defined on noncompact fuzzy number space and its completeness and separability are proved. Secondly, it turns out that some different concepts of noncompact fuzzy random variables are unified as the measurability is concerned. Furthermore, the variance and covariance of noncompact fuzzy random variables are introduced, and their properties are discussed. Finally, the fuzzy stochastic processes are also touched upon.
The paper is organized as follows:
Chapter 1 presents a summary of fuzzy random variables. Topics include a brief introduction of the idea of applying fuzzy random variables, an introduction of literature on fuzzy random variables and the problems considered in this thesis.
Noncompact fuzzy number space is introduced in Chapter 2. A generalized metric is defined on the noncompact fuzzy number set. Some properties and theorems about noncompact fuzzy numbers are discussed. Compared with the separability and incompleteness of compact fuzzy number space, the generalized metric defined here is proved to be complete and separable.
Noncompact fuzzy random variables are introduced in Chapter 3. We have compared some different concepts of noncompact fuzzy random variables. As the result, it will be shown that these concepts are unified as the measurability is concerned. The study extends some results of "A unified approach to fuzzy random variables (Volker Kratschmer, Fuzzy Sets and
Systems, 2001, 123, 1-9)". The concepts and properties of noncompact p-order fuzzy random variables are presented as extension of the p-order random vector. Finally, we defined the expectation of noncompact fuzzy random variables by a new method.
Noncompact fuzzy stochastic processes and their applications are put in Chapter 4. Some properties of noncompact 2-order fuzzy random variables and their applications are discussed. The results extend the study of "The variance and covariance of fuzzy random variables and their applications
(Feng Yuhu.et al, Fuzzy Sets and Systems, 2001, 120, 487-497.) ". In addition, we get a result about the right-closed-right-continuous fuzzy martingale.
引文
[1] LA Zadeh, Fuzzy Sets, Inform at Control 8(1965) 338-353.
[2] 文普寅，吴孟达，模糊理论及其应用，1998，国防科技大学出版社，1-6．
[3] R Kruse, K D Meyer, Statistics with vague data, Reidel, Dordrecht, 1987.
[4] H Kwakernaak, Fuzzy random variables, Definition and theorems, Inform Sci 15(1978)1-29.
[5] M L Puri and D A Ralescu, Fuzzy random variables, J Math Anal Appl 114(1986) 409-422.
[6] E P Klement, M L Puri, D A Ralescu, Limit theorems for fuzzy random variables, Proc Roy Soc London A 407( 1986) 171-182.
[7] P Diamond, P Kloeden. Metric spaces of fuzzy sets, Fuzzy sets and systems, 35(1990)241-249
[8] P Diamond, P Kloeden, Metric spaces of fuzzy sets, Singapore :World Scientific, 1994.
[9] V Krtschmer. A unified approach to fuzzy random variables. Fuzzy sets and systems 123(1) (2001) 1-9.
[10] Y Feng, L Hu, H Shu, The variance and covariance of fuzzy random variables and their applications, Fuzzy sets and systems 120(2001)487-497.
[11] R Korner, On the variance of fuzzy random variables, Fuzzy Sets and Systems 92(1997)83-93.
[12] Y Feng, Fuzzy random variables and fuzzy stochastic processes,东华大学自编讲义．
[13] Y Feng, Gaussian fuzzy random variables, Fuzzy Sets and Systems 111(2000)325-330.
[14] Y Feng, Sums of independent fuzzy random variables, Fuzzy sets and systems 123 (1)(2001) 10-18.
[15] Y Ogura, S Li, Separability for graph convergence of fuzzy-valued random variables, Fuzzy sets and systems 123 (1)(2001) 19-27.
[16] Y Feng, An approach to generalize laws of large numbers for fuzzy random variables, Fuzzy sets and systems 128(2002)237-245.
[17] H Inour;A strong law of large numbers for fuzzy random sets, Fuzzy sets and systems 41(1991)285-291.
[18] Y Feng, Fuzzy-valued mappings with variation, fuzzy-valued measures and fuzzy valued lebegue-stieltjes integrals, Fuzzy sets and systems 121(2001)225-234.
[19] Y Feng, Mean-square integral and differential of fuzzy stochastic processes, Fuzzy Sets and Systems 102(1999) 271-280
[20] Y Feng, Mean-square Riemann-Stieltjes integrals of fuzzy stochastic processes and their applications, Fuzzy Sets and Systems 110(2000) 27-41.
[21] C Chen, W Chen, Analysis and design of a stable fuzzy control systems, Fuzzy Sets and Systems 96,1998,21-35.
[22] O Kaleva, Fuzzy differential equation, Fuzzy sets and systems 24 (1987) 301-317.
[23] O Kaleva, The Cauchy problem for fuzzy differential equations, Fuzzy Sets and Systems 35(1990) 389-396.
[24] Y Liu, Z Qiao and G.Wang, Fuzzy random reliability of structures based on fuzzy random variables, Fuzzy Sets and Systems 86(1997) 345-355.
[25] M Wu, Linear fuzzy differential equation systems on R1, J Fuzzy Systems Math 2(1)(1988)51-56 (in Chinese) J of Math Analysis and Applications 202(1996)629-644.
[26] T Zhang, Stable adaptive fuzzy sliding mode control of interconnected systems, Fuzzy Sets and Systems 122(2001)5-19.
[27] J Zhang, R Li and P Zhang, Stability analysis and systematic design of fuzzy control systems, Fuzzy Sets and Systems 120(2001)65-72.
[28] Y Feng, On the convergence of fuzzy martingales, Fuzzy Sets and Systems 130(2002) 67-73.
[29] Y Feng, Decomposition theorems for fuzzy supermartingales and submartingales, Fuzzy sets and systems 116(2000)225-235.
[30] S Li,Y Pgura, Convergence of set-valued and fuzzy-valued martingales, Fuzzy sets and systems 101(1999)453-461.
[31] M L Puri,D A Ralescu, Convergence theorems for fuzzy martingales, J Math Anal Appal 160(1991)107-122.
[32] Y Feng, Convergence theorems for fuzzy random variables and fuzzy martingales,Fuzzy sets and systems 103 (1999)435 -441.
[33] G Feng, S G Cao, N W Rees and C K Chak, Design of fuzzy control systems with guaranteed stability, Fuzzy Sets and Systems 85(1997) 1-10.
[34] M Friedman, M Ma, A Kandel, Fuzzy Linear systems, Fuzzy Sets and Systems 96(1998) 201-209.
[35] L Hu, R Wu, S Shao, Analysis of dynamical systems whose inputs are fuzzy stochastic process, Fuzzy sets and systems 129(2002)111-128.
[36] Jun Yoneyama, Masahiro Nishikawa, Hitoshi Katayama and Akira Ichikawa, Output stabilization of Takagi-Sugeno fuzzy systems, Fuzzy Sets and Systems 111 (2000) 253-266.
[37] G Debreu, Integration of correspondences, in: Proceedings of the Fifth Berkely Symposium on Mathematical Statistics and Probability, Vol Ⅱ, Part Ⅰ. Berkeley, Los Angeles: University of California Press, 1967, 351-372.
[38] A Kandel, Fuzzy dynamical systems and the nature of their solutions, in: P P Wang, S K Chang (Eds), Fuzzy Sets: Theory and Application to Policy Analysis and Information Systems, Plenum Press, New York,
1980, 93-122.
[39] V Lakshmikantham, Uncertain systems and fuzzy differential equations, J of Math Analysis and Applications 251 (2000)805-817.
[40] J H Kim, Y S Yun, On fuzzy stochastic differential equations(to appear)
[41] Y Feng, Metric on noncompact fuzzy number space (?)~n,J Dong Hua University(Eng Ed)(即将发表)．
[42] V Krtschmer. Some complete merics on spaces of fuzzy subsets. Fuzzy sets and systems 130(3)(2002) 357-365.
[43] [美]J L凯莱，一般拓扑学，1982，科学出版社．
[44] K R Parthasarathy, Probability Measures on Metric Spaces, Academic Press, New York, 1975.
[45] 张文修，汪振鹏，高勇，集值随机变量，1996，科学出版社．
[46] M Stojakovic, Representation of fuzzy-valued mappings by the sequence of single-valued mappings, Fuzzy Sets and Systems 98(1998) 375-381.
[2] 文普寅，吴孟达，模糊理论及其应用，1998，国防科技大学出版社，1-6．
[3] R Kruse, K D Meyer, Statistics with vague data, Reidel, Dordrecht, 1987.
[4] H Kwakernaak, Fuzzy random variables, Definition and theorems, Inform Sci 15(1978)1-29.
[5] M L Puri and D A Ralescu, Fuzzy random variables, J Math Anal Appl 114(1986) 409-422.
[6] E P Klement, M L Puri, D A Ralescu, Limit theorems for fuzzy random variables, Proc Roy Soc London A 407( 1986) 171-182.
[7] P Diamond, P Kloeden. Metric spaces of fuzzy sets, Fuzzy sets and systems, 35(1990)241-249
[8] P Diamond, P Kloeden, Metric spaces of fuzzy sets, Singapore :World Scientific, 1994.
[9] V Krtschmer. A unified approach to fuzzy random variables. Fuzzy sets and systems 123(1) (2001) 1-9.
[10] Y Feng, L Hu, H Shu, The variance and covariance of fuzzy random variables and their applications, Fuzzy sets and systems 120(2001)487-497.
[11] R Korner, On the variance of fuzzy random variables, Fuzzy Sets and Systems 92(1997)83-93.
[12] Y Feng, Fuzzy random variables and fuzzy stochastic processes,东华大学自编讲义．
[13] Y Feng, Gaussian fuzzy random variables, Fuzzy Sets and Systems 111(2000)325-330.
[14] Y Feng, Sums of independent fuzzy random variables, Fuzzy sets and systems 123 (1)(2001) 10-18.
[15] Y Ogura, S Li, Separability for graph convergence of fuzzy-valued random variables, Fuzzy sets and systems 123 (1)(2001) 19-27.
[16] Y Feng, An approach to generalize laws of large numbers for fuzzy random variables, Fuzzy sets and systems 128(2002)237-245.
[17] H Inour;A strong law of large numbers for fuzzy random sets, Fuzzy sets and systems 41(1991)285-291.
[18] Y Feng, Fuzzy-valued mappings with variation, fuzzy-valued measures and fuzzy valued lebegue-stieltjes integrals, Fuzzy sets and systems 121(2001)225-234.
[19] Y Feng, Mean-square integral and differential of fuzzy stochastic processes, Fuzzy Sets and Systems 102(1999) 271-280
[20] Y Feng, Mean-square Riemann-Stieltjes integrals of fuzzy stochastic processes and their applications, Fuzzy Sets and Systems 110(2000) 27-41.
[21] C Chen, W Chen, Analysis and design of a stable fuzzy control systems, Fuzzy Sets and Systems 96,1998,21-35.
[22] O Kaleva, Fuzzy differential equation, Fuzzy sets and systems 24 (1987) 301-317.
[23] O Kaleva, The Cauchy problem for fuzzy differential equations, Fuzzy Sets and Systems 35(1990) 389-396.
[24] Y Liu, Z Qiao and G.Wang, Fuzzy random reliability of structures based on fuzzy random variables, Fuzzy Sets and Systems 86(1997) 345-355.
[25] M Wu, Linear fuzzy differential equation systems on R1, J Fuzzy Systems Math 2(1)(1988)51-56 (in Chinese) J of Math Analysis and Applications 202(1996)629-644.
[26] T Zhang, Stable adaptive fuzzy sliding mode control of interconnected systems, Fuzzy Sets and Systems 122(2001)5-19.
[27] J Zhang, R Li and P Zhang, Stability analysis and systematic design of fuzzy control systems, Fuzzy Sets and Systems 120(2001)65-72.
[28] Y Feng, On the convergence of fuzzy martingales, Fuzzy Sets and Systems 130(2002) 67-73.
[29] Y Feng, Decomposition theorems for fuzzy supermartingales and submartingales, Fuzzy sets and systems 116(2000)225-235.
[30] S Li,Y Pgura, Convergence of set-valued and fuzzy-valued martingales, Fuzzy sets and systems 101(1999)453-461.
[31] M L Puri,D A Ralescu, Convergence theorems for fuzzy martingales, J Math Anal Appal 160(1991)107-122.
[32] Y Feng, Convergence theorems for fuzzy random variables and fuzzy martingales,Fuzzy sets and systems 103 (1999)435 -441.
[33] G Feng, S G Cao, N W Rees and C K Chak, Design of fuzzy control systems with guaranteed stability, Fuzzy Sets and Systems 85(1997) 1-10.
[34] M Friedman, M Ma, A Kandel, Fuzzy Linear systems, Fuzzy Sets and Systems 96(1998) 201-209.
[35] L Hu, R Wu, S Shao, Analysis of dynamical systems whose inputs are fuzzy stochastic process, Fuzzy sets and systems 129(2002)111-128.
[36] Jun Yoneyama, Masahiro Nishikawa, Hitoshi Katayama and Akira Ichikawa, Output stabilization of Takagi-Sugeno fuzzy systems, Fuzzy Sets and Systems 111 (2000) 253-266.
[37] G Debreu, Integration of correspondences, in: Proceedings of the Fifth Berkely Symposium on Mathematical Statistics and Probability, Vol Ⅱ, Part Ⅰ. Berkeley, Los Angeles: University of California Press, 1967, 351-372.
[38] A Kandel, Fuzzy dynamical systems and the nature of their solutions, in: P P Wang, S K Chang (Eds), Fuzzy Sets: Theory and Application to Policy Analysis and Information Systems, Plenum Press, New York,
1980, 93-122.
[39] V Lakshmikantham, Uncertain systems and fuzzy differential equations, J of Math Analysis and Applications 251 (2000)805-817.
[40] J H Kim, Y S Yun, On fuzzy stochastic differential equations(to appear)
[41] Y Feng, Metric on noncompact fuzzy number space (?)~n,J Dong Hua University(Eng Ed)(即将发表)．
[42] V Krtschmer. Some complete merics on spaces of fuzzy subsets. Fuzzy sets and systems 130(3)(2002) 357-365.
[43] [美]J L凯莱，一般拓扑学，1982，科学出版社．
[44] K R Parthasarathy, Probability Measures on Metric Spaces, Academic Press, New York, 1975.
[45] 张文修，汪振鹏，高勇，集值随机变量，1996，科学出版社．
[46] M Stojakovic, Representation of fuzzy-valued mappings by the sequence of single-valued mappings, Fuzzy Sets and Systems 98(1998) 375-381.