不确定更新过程及其积分理论
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摘要
不确定理论是处理人的信度的一个数学系统.不确定过程用来描述随时间变化的不确定现象,本质上它是一列不确定变量.有一类轨道不连续的不确定过程称为不确定更新过程,它用来描述一类变化过程不连续的不确定现象.在不确定更新过程的基础上,本论文提出了不确定交错更新过程用来描述交替工作和休息的不确定系统,计算了它的平均工作率的不确定分布,并证明了基本更新定理,完善了不确定更新理论.
基于不确定更新过程,本论文建立了一套积分理论,用来处理不确定过程关于不确定更新过程的积分和微分.同时,本论文证明了这种积分满足关于被积项的线性性和关于积分区间的可加性,并给出了不确定更新过程的积分理论的基本定理.不确定更新过程的积分理论扩展了不确定积分理论的研究内容,为研究带跳不确定微分方程奠定了基础.
为了描述不连续变化的不确定现象所服从的变化规律,本论文提出了由不确定更新过程驱动的微分方程,即带跳不确定微分方程.本论文给出了两类带跳不确定微分方程的解析解,并证明了带跳不确定微分方程的解的存在唯一性定理.此外,本论文还定义了带跳不确定微分方程在不确定测度意义下的稳定性,给出了带跳不确定微分方程稳定的充分条件.本论文为研究不确定环境下带跳的金融市场及最优控制等提供了理论依据,使不确定微分方程能够更好的指导实践.
本文的创新点主要有：
·定义了不确定交错更新过程,给出了平均工作率的不确定分布,证明了交错更新定理；
·建立了不确定过程关于不确定更新过程的积分理论,研究了该积分的一些数学性质,并给出了它的基本定理；
·定义了带跳不确定微分方程,给出了两类方程的解析解,并证明了方程的解的存在唯一性和稳定性定理.
Uncertainty theory is a branch of axiomatic mathematics to deal with human’sbelief degree. Uncertain process, aiming to describe the evolution of uncertain phe-nomena, is essentially a sequence of uncertain variables. Uncertain renewal process isa type of sample-discontinuous uncertain process, and it is used to model discontinu-ously varying uncertain phenomena. Based on uncertain renewal process, this thesisproposes a new type of uncertain process called uncertain alternating renewal processto describe an uncertain system which works and rests alternately. The uncertainty dis-tribution of the average working rate is given, and the alternating renewal theorem isproved. As a result, the uncertain renewal theory is expanded.
Based on uncertain renewal process, this thesis builds a new theory of uncertaincalculus to deal with the integral and diferential of an uncertain process with respectto uncertain renewal process. The integral is proved to meet with the linearity on theintegrand and the additivity on the bounds. In addition, the fundamental theorem of un-certain calculus with uncertain renewal process is verified, which gives the diferentialof a function of uncertain process with respect to uncertain renewal process. Uncertaincalculus with uncertain renewal process, which extends the area of uncertain calculustheory, is the basis to study uncertain diferential equation with jumps.
In order to describe the rule that a discontinuously varying uncertain phenomenonobeys, this thesis proposes a type of diferential equation driven by uncertain renewalprocess, i.e., uncertain diferential equation with jumps. It gives analytic solutions fortwo types of uncertain diferential equations with jumps. Besides, it gives an existenceand uniqueness theorem for the proposed diferential equation. In addition, this thesisproposes a definition of stability in the sense of uncertain measure for uncertain difer-ential equation with jumps, and gives a sufcient condition for the diferential equationbeing stable. These results provide a theoretical basis for further research in many ar-eas such as uncertain financial market and uncertain optimal control with jumps. As a result, uncertain diferential equation will do a better job in practice.
The contributions of this thesis are:
It proposes a definition of uncertain alternating renewal process, and gives anuncertainty distribution of average working rate. Besides, it proves an alternatingrenewal theorem.
It builds uncertain calculus with respect to uncertain renewal process. Someproperties of the integral are investigated, and the fundamental theorem is veri-fied.
It introduces a concept of uncertain diferential equation with jumps, and givesanalytic solutions for two types of the proposed diferential equations. In ad-dition, it gives sufcient conditions for an uncertain diferential equation withjumps to have a unique solution and to be stable.
基于不确定更新过程,本论文建立了一套积分理论,用来处理不确定过程关于不确定更新过程的积分和微分.同时,本论文证明了这种积分满足关于被积项的线性性和关于积分区间的可加性,并给出了不确定更新过程的积分理论的基本定理.不确定更新过程的积分理论扩展了不确定积分理论的研究内容,为研究带跳不确定微分方程奠定了基础.
为了描述不连续变化的不确定现象所服从的变化规律,本论文提出了由不确定更新过程驱动的微分方程,即带跳不确定微分方程.本论文给出了两类带跳不确定微分方程的解析解,并证明了带跳不确定微分方程的解的存在唯一性定理.此外,本论文还定义了带跳不确定微分方程在不确定测度意义下的稳定性,给出了带跳不确定微分方程稳定的充分条件.本论文为研究不确定环境下带跳的金融市场及最优控制等提供了理论依据,使不确定微分方程能够更好的指导实践.
本文的创新点主要有：
·定义了不确定交错更新过程,给出了平均工作率的不确定分布,证明了交错更新定理；
·建立了不确定过程关于不确定更新过程的积分理论,研究了该积分的一些数学性质,并给出了它的基本定理；
·定义了带跳不确定微分方程,给出了两类方程的解析解,并证明了方程的解的存在唯一性和稳定性定理.
Uncertainty theory is a branch of axiomatic mathematics to deal with human’sbelief degree. Uncertain process, aiming to describe the evolution of uncertain phe-nomena, is essentially a sequence of uncertain variables. Uncertain renewal process isa type of sample-discontinuous uncertain process, and it is used to model discontinu-ously varying uncertain phenomena. Based on uncertain renewal process, this thesisproposes a new type of uncertain process called uncertain alternating renewal processto describe an uncertain system which works and rests alternately. The uncertainty dis-tribution of the average working rate is given, and the alternating renewal theorem isproved. As a result, the uncertain renewal theory is expanded.
Based on uncertain renewal process, this thesis builds a new theory of uncertaincalculus to deal with the integral and diferential of an uncertain process with respectto uncertain renewal process. The integral is proved to meet with the linearity on theintegrand and the additivity on the bounds. In addition, the fundamental theorem of un-certain calculus with uncertain renewal process is verified, which gives the diferentialof a function of uncertain process with respect to uncertain renewal process. Uncertaincalculus with uncertain renewal process, which extends the area of uncertain calculustheory, is the basis to study uncertain diferential equation with jumps.
In order to describe the rule that a discontinuously varying uncertain phenomenonobeys, this thesis proposes a type of diferential equation driven by uncertain renewalprocess, i.e., uncertain diferential equation with jumps. It gives analytic solutions fortwo types of uncertain diferential equations with jumps. Besides, it gives an existenceand uniqueness theorem for the proposed diferential equation. In addition, this thesisproposes a definition of stability in the sense of uncertain measure for uncertain difer-ential equation with jumps, and gives a sufcient condition for the diferential equationbeing stable. These results provide a theoretical basis for further research in many ar-eas such as uncertain financial market and uncertain optimal control with jumps. As a result, uncertain diferential equation will do a better job in practice.
The contributions of this thesis are:
It proposes a definition of uncertain alternating renewal process, and gives anuncertainty distribution of average working rate. Besides, it proves an alternatingrenewal theorem.
It builds uncertain calculus with respect to uncertain renewal process. Someproperties of the integral are investigated, and the fundamental theorem is veri-fied.
It introduces a concept of uncertain diferential equation with jumps, and givesanalytic solutions for two types of the proposed diferential equations. In ad-dition, it gives sufcient conditions for an uncertain diferential equation withjumps to have a unique solution and to be stable.
引文
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[5]Ito K. On Stochastic Differential Equations. Memoirs of the American Mathematical Soci-ety,1951,4:1-51.
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[13](?)ksendal B, Sulem A. Applied Stochastic Control of Jump Diffusions.2nd ed., Berlin: Springer-Verlag,2007.
[14]Hanson F B. Applied Stochastic Processes and Control for Jump-Diffusions:Modelling, Analysis, and Computation. Philadelphia:Socity for Industrial and Applied Mathematics,2007.
[15]Jabod J, Protter P. Time Reversal on Levy Processes. The Annals of Probability,1988,16(2):620-641.
[16]Bertoin J. Levy Processes. Cambridge:Cambridge University Press,1996.
[17]Applebaum D. Levy Processes and Stochatic Calculus. Cambridge:Cambridge University Press,2004.
[18]Liao M. Levy Processes in Lie Groups. Cambridge:Cambridge University Press,2004.
[19]Doob J L. Stochastic Processes. New York:John Wiley&Sons,1953.
[20]Kunita H, Watanabe S. On Square Integrable Martingales. Nagoya Mathematical Journal,1967,30:209-245.
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[30]Gao X. Some Properties of Continuous Uncertain Measure. International Journal of Uncer-tainty, Fuzziness and Knowledge-Based Systems,2009,17(3):419-426.
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[44]Liu B. Theory and Practice of Uncertain Programming.2nd ed., Berlin:Springer-Verlag,2009.
[45]Gao Y. Shortest Path Problem with Uncertain Arc Lengths. Computers and Mathematics with Applications,2011,62(6):2591-2600.
[46]Gao Y. Uncertain Models for Single Facility Location Problems on Networks. Applied Mathematical Modelling,2012,36(6):2592-2599.
[47]Sheng Y, Yao K. Fixed Charge Transportation Problem in Uncertain Environment. Industrial Engineering and Management Systems,2012,11(2):183-187.
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[50]Chen X, Ralescu D A. B-Spline Method of Uncertain Statistics with Application to Esti-mating Travel Distance. Journal of Uncertain Systems,2012,6(4):256-262.
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[2]Ito K. Stochastic Integral. Proceedings of the Japan Academy,1944,20(8):519-524.
[3]Ito K. On a Stochastic Integral Equation. Proceedings of the Japan Academy,1946,22(2):32-35.
[4]Ito K. Stochastic Differential Equations in a Differentiable Manifold. Nagoya Mathematical Journal,1950,1:35-47.
[5]Ito K. On Stochastic Differential Equations. Memoirs of the American Mathematical Soci-ety,1951,4:1-51.
[6]Kalman R E, Bucy R S. New Results in Linear Filtering and Prediction Theory. Journal of Basic Engineering,1961,83:95-108.
[7]Black F, Scholes M. The Pricing of Options and Corporate Liabilities. Journal of Political Economy,1973,81:637-654.
[8](?)ksendal B. Stochastic Differential Equations:An Introduction with Applications.6th ed., Berlin:Springer-Verlag,2003.
[9]Protter P. Stochastic Integration and Differential Equations.2nd ed., Berlin:Springer-Verlag,2004.
[10]胡适耕,黄乘明,吴付科.随机微分方程.北京：科学出版社,2008.
[11]龚光鲁.随机微分方程及其应用概要.北京：清华大学出版社,2008.
[12]Bichteler K. Stochastic Integration with Jumps. Cambridge:Cambridge University Press,2002.
[13](?)ksendal B, Sulem A. Applied Stochastic Control of Jump Diffusions.2nd ed., Berlin: Springer-Verlag,2007.
[14]Hanson F B. Applied Stochastic Processes and Control for Jump-Diffusions:Modelling, Analysis, and Computation. Philadelphia:Socity for Industrial and Applied Mathematics,2007.
[15]Jabod J, Protter P. Time Reversal on Levy Processes. The Annals of Probability,1988,16(2):620-641.
[16]Bertoin J. Levy Processes. Cambridge:Cambridge University Press,1996.
[17]Applebaum D. Levy Processes and Stochatic Calculus. Cambridge:Cambridge University Press,2004.
[18]Liao M. Levy Processes in Lie Groups. Cambridge:Cambridge University Press,2004.
[19]Doob J L. Stochastic Processes. New York:John Wiley&Sons,1953.
[20]Kunita H, Watanabe S. On Square Integrable Martingales. Nagoya Mathematical Journal,1967,30:209-245.
[21]Karatzas I, Shreve S. Brownian Motion and Stochastic Calculus.2nd ed., Berlin:Springer-Verlag,1991.
[22]Revuz D, Yor M. Continuous Martingales and Brownian Motion.3rd ed., Berlin:Springer-Verlag,1999.
[23]何声武,汪嘉冈,严加安.半鞅与随机分析.北京：科学出版社,1995.
[24]严加安.随机分析选讲.北京：科学出版社,1997.
[25]Kahneman D, Tversky A. Prospect Theory:An Analysis of Decisions under Risk. Econo-metrica,1979,47:263-291.
[26]Liu B. Why is There a Need for Uncertainty Theory. Journal of Uncertain Systems,2011,5(1):3-20.
[27]Liu B. Uncertainty Theory.2nd ed., Berlin:Springer-Verlag,2007.
[28]Liu B. Uncertainty Theory:A Branch of Mathematics for Modeling Human Uncertainty. Berlin:Springer-Verlag,2010.
[29]Liu B. Some Research Problems in Uncertainty Theory. Journal of Uncertain Systems,2009,3(1):3-10.
[30]Gao X. Some Properties of Continuous Uncertain Measure. International Journal of Uncer-tainty, Fuzziness and Knowledge-Based Systems,2009,17(3):419-426.
[31]高欣.不确定测度及其应用[博士学位论文].北京：清华大学,2009.
[32]Zhang Z. Some Discussions on Uncertain Measure. Fuzzy Optimization and Decision Making,2011,10(1):31-43.
[33]彭子雄.复不确定变量[博士学位论文].北京：清华大学,2012.
[34]Liu Y H, Ha M. Expected Value of Function of Uncertain Variables. Journal of Uncertain Systems,2010,4(3):181-186.
[35]You C. Some Convergence Theorems of Uncertain Sequences. Mathematical and Computer Modelling,49(3):482-487.
[36]Tian J. Inequalities and Mathematical Properties of Uncertain Variables. Fuzzy Optimiza-tion and Decision Making,2011,10(4):357-368.
[37]Chen X, Dai W Maximum Entropy Principle for Uncertain Variables. International Journal of Fuzzy Systems,2011,13(3):232-236.
[38]Chen X, Samarjit K, Ralescu D A. Cross-Entropy Measure of Uncertain Variables. Infor-mation Sciences,2012,201:53-60.
[39]Dai W, Chen X. Entropy of Function of Uncertain Variables. Mathematical and Computer Modelling,2012,55(3):754-760.
[40]戴韡.不确定理论中的极大熵原理[博士学位论文].北京：清华大学,2010.
[41]Liu B. Uncertain Risk Analysis and Uncertain Reliability Analysis. Journal of Uncertain Systems,2010,4(3):163-170.
[42]Liu J. Uncertain Comprehensive Evaluation Method. Journal of Information&Computa-tional Science,2011,8(2):336-344.
[43]Liu W. Reliability Analysis of Redundant System with Uncertain Lifetimes. Information: An International Interdisciplinary Journal,2013,16(2):881-888.
[44]Liu B. Theory and Practice of Uncertain Programming.2nd ed., Berlin:Springer-Verlag,2009.
[45]Gao Y. Shortest Path Problem with Uncertain Arc Lengths. Computers and Mathematics with Applications,2011,62(6):2591-2600.
[46]Gao Y. Uncertain Models for Single Facility Location Problems on Networks. Applied Mathematical Modelling,2012,36(6):2592-2599.
[47]Sheng Y, Yao K. Fixed Charge Transportation Problem in Uncertain Environment. Industrial Engineering and Management Systems,2012,11(2):183-187.
[48]Wang X, Gao Z, Guo H. Uncertain Hypothesis Testing for Two Experts'Empirical Data. Mathematical and Computer Modelling,2012,55:1478-1482.
[49]Wang X, Gao Z, Guo H. Delphi Method for Estimating Uncertainty Distributions. Informa-tion:An International Interdisciplinary Journal,2012,15(2):449-460.
[50]Chen X, Ralescu D A. B-Spline Method of Uncertain Statistics with Application to Esti-mating Travel Distance. Journal of Uncertain Systems,2012,6(4):256-262.
[51]Liu B. Uncertain Set Theory and Uncertain Inference Rule with Application to Uncertain Control. Journal of Uncertain Systems,2010,4(2):83-98.
[52]Gao X, Gao Y, Ralescu D A. On Liu's Inference Rule for Uncertain Systems. International Journal of Uncertainty, Fuzziness and Knowledge-Based Systems,2010,18(1):1-11.
[53]Gao Y. Uncertain Inference Control for Balancing an Inverted Pendulum. Fuzzy Optimiza-tion and Decision Making,2012,11(4):481-492.
[54]Li X, Liu B. Hybrid Logic and Uncertain Logic. Journal of Uncertain Systems,2009,3(2):83-94.
[55]Liu B. Uncertain Entailment and Modus Ponens in the Framework of Uncertain Logic. Journal of Uncertain Systems,2009,3(4):243-251.
[56]Chen X, Ralescu D A. A Note on Truth Value in Uncertain Logic. Expert Systems with Applications,2011,38:15582-15586.
[57]Liu B. Uncertain Logic for Modeling Human Language. Journal of Uncertain Systems,2011,5(1):3-20.
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