JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE) ›› 2019, Vol. 54 ›› Issue (8): 1-13.doi: 10.6040/j.issn.1671-9352.0.2019.235

    Next Articles

Itô-Henstock integration of the fuzzy stochastic process

GONG Zeng-tai, SU Ai   

  1. College of Mathematics and Statistics, Northwest Normal University, Lanzhou 730070, Gansu, China
  • Online:2019-08-20 Published:2019-07-03

PDF (PC)

709

Abstract: The fuzzy Itô-Henstock integral and the fuzzy Itô-McShane integrals for adapted fuzzy stochastic process with respect to a Brownian motion are defined and their properties are discussed. In addition, the characterization theorems are given and their interrelation of between the fuzzy Itô-Henstock integral and the fuzzy Itô-McShane integral is investigated. The result shows that the fuzzy Itô-Henstock integral is equivalent to the fuzzy Itô-McShane integral when its primitive of fuzzy Itô-Henstock integral is Itô absolutely continuous.

Key words: fuzzy number, fuzzy stochastic variable, Brownian motion, fuzzy stochastic process, fuzzy Itô, integral

CLC Number: 

  • O172.2
[1] IT(＾overO)K. Stochastic integral[J]. Proc Imp Acad Tokyo, 1944, 20(8):519-524.
[2] KWAKERNAAK H. Fuzzy random variables: I. Definitions and theorems[J]. Information Sciences, 1978, 15(1):1-29.
[3] PURI M L, RALESCU D A. Fuzzy random variables[J]. Journal of Mathematical Analysis and Applications, 1986, 114(2):409-422.
[4] KIM B K, KIM J H. Stochastic integrals of set-valued processes and fuzzy processes[J]. Journal of Mathematical Analysis and Applications, 1999, 236(2):480-502.
[5] JUNG E J, KIM J H. On set-valued stochastic integrals[J]. Stochastic Analysis and Applications, 2003, 21(2):401-418.
[6] KISIELEWICZ M. Existence theorem for nonconvex stochastic inclusions[J]. Journal of Application and Mathematical Stochastic Analysis, 1994, 7(2):151-159.
[7] KISIELEWICZ M. Viability theorems for stochastic inclusions[J]. Discussiones Mathematicae Differential Inclusions, Control and Optimization, 1995, 15(1):61-74.
[8] LI Shoumei, REN Aihong. Representation theorems, set-valued and fuzzy set-valued Itô-integral[J]. Fuzzy Sets and Systems, 2007, 158(9):949-962.
[9] MCSHANE E J. Stochastic calculus and stochastic models[M]. New York: Academic Press, 1974.
[10] PROTTER P. A comparison of stochastic integrals[J]. Annals of Probability, 1979, 7(2):276-289.
[11] XU J G, LEE P Y. Stochastic integrals of Itô and Henstock[J]. Real Analysis Exchange, 1992/1993, 18:352-366.
[12] CHEW T S, TAY J Y, TOH T L. The non-uniform Riemann approach to Itôs integral[J]. Real Analysis Exchange, 2001/2002, 27(2):495-514.
[13] 黄志远. 随机分析学基础[M]. 2版. 北京: 科学出版社, 2001. HUANG Zhiyuan. Stochastic analytical fundation[M]. 2nd ed. Beijing: Science Press, 2001.
[14] 吴从炘, 马明. 模糊分析学基础[M]. 北京: 国防工业出版社, 1991. WU Congxin, MA Ming. Fuzzy analytical foundation[M]. Beijing: Defense Industry Press, 1991.
[15] FENG Yuhu. Mean-squares Riemann-Stieltjes integrals of fuzzy stochastic processes and their applications[J]. Fuzzy Sets and Systems, 2000, 110(1):27-41.
[16] WU Congxin, GONG Zengtai. On Henstock integral of fuzzy number-valued function(I)[J]. Fuzzy Sets and Systems, 2001, 120(3):523-532.
[17] HENSTOCK R. The general theory of integration[M]. Oxford: Oxford University Press, 1991.
[18] 李艳红, 王贵君. 广义模糊值Choquet积分的强序连续与伪S性[J]. 山东大学学报(理学版), 2008, 43(4):76-80. LI Yanhong, WANG Guijun. Strongly order continuity and pseudo-S-property of generalized fuzzy valued Choquet integrals[J]. Journal of Shandong University(Natural Science), 2008, 43(4):76-80.
[1] REN Jian-long. Reconstruction of unknown surface heat flux from an internal temperature history [J]. JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE), 2019, 54(9): 83-90.
[2] QI Ting-ting, ZHANG Zhen-fu, LIU Yan-sheng. Existence of positive solutions for fractional differential system with coupled integral boundary conditions [J]. JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE), 2019, 54(2): 71-78.
[3] TAO Shuang-ping, GAO Rong. Estimates of multilinear fractional integrals and maximal operators on weighted Morrey spaces [J]. JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE), 2018, 53(6): 30-37.
[4] XIN Yin-ping, TAO Shuang-ping. Boundedness of Marcinkiewicz integrals operators with variable kernels on Herz-type Hardy spaces with variable exponent [J]. JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE), 2018, 53(6): 38-43.
[5] CHENG Cheng, ZOU Shi-jia. Irreducible splitting trace module of a class of Hopf algebras [J]. JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE), 2018, 53(4): 11-15.
[6] GONG Zeng-tai, GAO Han. Preinvexity of n-dimensional fuzzy number-valued functions [J]. JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE), 2018, 53(10): 72-81.
[7] GONG Zeng-tai, KOU Xu-yang. Representation of Choquet integral of the set-valued functions with respect to fuzzy measures and the characteristic of its primitive [J]. JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE), 2017, 52(8): 1-9.
[8] FENG Hai-xing, ZHAI Cheng-bo. Multiple positive solutions of a system of high order nonlinear fractional differential equations [J]. JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE), 2017, 52(12): 48-57.
[9] CUI Jing, LIANG Qiu-ju. Existence and controllability of nonlocal stochastcic integro-differential equations driven by fractional Brownian motion [J]. JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE), 2017, 52(12): 81-88.
[10] YAO Jun-qing, ZHAO Kai. Commutators of fractional integrals on Herz-Morrey spaces with variable exponent [J]. JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE), 2017, 52(11): 100-105.
[11] . Application of fuzzy differential transform method for solving fuzzy integral differential equation [J]. JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE), 2017, 52(10): 42-49.
[12] KONG Yi-ting, WANG Tong-ke. The steepest descent method for Fourier integrals involving algebraic and logarithmic singular factors [J]. JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE), 2017, 52(10): 50-55.
[13] SU Xiao-feng, JIA Mei, LI Meng-meng. Existence of solution for fractional differential equation integral boundary value problem at resonance [J]. JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE), 2016, 51(8): 66-73.
[14] WANG Jie, QU Meng, SHU Li-sheng. Boundedness of the Littlewood-Paley operators and cummutators on the Herz spaces with variable exponents [J]. JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE), 2016, 51(4): 9-18.
[15] CUI Jian-bin, JI An-zhao, LU Hong-jiang, WANG Yu-feng, HE Jiang-yi, XU Tai. Numerical solution of Schwarz Christoffel transform [J]. JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE), 2016, 51(4): 104-111.
Viewed
Full text


Abstract

Cited
  Shared   
  Discussed   
[1] TANG Xiao-hong1, HU Wen-xiao2*, WEI Yan-feng2, JIANG Xi-long2, ZHANG Jing-ying2, SHAO Xue-dong3. Screening and biological characteristics studies of wide wine-making yeasts[J]. JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE), 2014, 49(03): 12 -17 .
[2] YI Chao-qun, LI Jian-ping, ZHU Cheng-wen. A kind of feature selection based on classification accuracy of SVM[J]. J4, 2010, 45(7): 119 -121 .
[3] SUN Liang-ji,JI Guo-xing . Jordan(α,β)-derivations and generalized Jordan(α,β)-derivations on upper triangular matrix algebras[J]. J4, 2007, 42(10): 100 -105 .
[4] ZHANG Jing-you, ZHANG Pei-ai, ZHONG Hai-ping. The application of evolutionary graph theory in the design of knowledge-based enterprises’ organization strucure[J]. J4, 2013, 48(1): 107 -110 .
[5] SHI Kai-quan. P-information law intelligent fusion and soft information #br# image intelligent generation[J]. JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE), 2014, 49(04): 1 -17 .
[6] WU Yun1,2, HAN Ya-die1. Existence of nontrivial solutions of nonlinear elliptic equation with critical potential[J]. J4, 2013, 48(4): 91 -94 .
[7] XIAO Hua . Continuous dependence of the solution of multidimensional reflected backward stochastic differential equations on the parameters[J]. J4, 2007, 42(2): 68 -71 .
[8] LIU Hong-hua . The alternating group iterative method for the dispersive equation[J]. J4, 2007, 42(1): 19 -23 .
[9] ZHANG Ji-long and YI Hong-xun . The uniqueness of function derivative that share one CM value[J]. J4, 2006, 41(1): 115 -119 .
[10] SONG Ying,ZHANG Xing-fang ,WANG Qing-ping . α-reverse triple I method in the parametric Kleene's system[J]. J4, 2007, 42(7): 45 -48 .
Copyright 2018 © Editorial office of JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE)
Supported by Beijing Magtech Co.Ltd