Document Type: Research Paper


College of Mathematics and Information Science, Hebei University, Baoding 071002, China


Uncertainty theory is a mathematical methodology for dealing with
non-determinate phenomena in nature. As we all know, uncertain
process and uncertain integral are important contents of uncertainty
theory, so it is necessary to have deep research. This paper
presents the definition and discusses some properties of strong
comonotonic uncertain process. Besides, some useful formulas of
uncertain integral such as nonnegativity, monotonicity, intermediate
results are studied.


[1] X. Chen and B. Liu, Existence and uniqueness theorem for uncertain differential equations,
Fuzzy Optimization and Decision Making, 9(1) (2010), 69{81.
[2] D. Dubois and H. Prade, Possibility Theory: An Approach to Computerized Processing of
Uncertainty, Plenum, New York, 1988.
[3] X. Gao, Some properties of continuous uncertain measure, International Journal of Uncer-
tainty, Fuzziness and Knowledge-Based System, 17(3) (2009), 419{426.
[4] M. Ha and X. Li, Choquet integral based on self-dual measure, Journal of Hebei University
(Natrual Science Edition), 28(2) (2008), 113{115.
[5] B. Liu and Y. K. Liu, Expected value of fuzzy variable and fuzzy expected value models, IEEE
Transactions on Fuzzy Systems, 10(4) (2002), 445{450.
[6] B. Liu, Uncertainty Theory, 2nd ed., Springer-Verlag, Berlin, 2007.
[7] B. Liu, Fuzzy process, hybrid process and uncertain process, Journal of Uncertain Systems,
2(1) (2008),3{16.
[8] B. Liu, Theory and Practice of Uncertain Programming, 2nd edn, Springer-Verlag, Berlin,
[9] B. Liu, Some research problems in uncertainty theory, Journal of Uncertain Systems, 3(1)
(2009), 3{10.
[10] B. Liu, Uncertainty Theory: A Branch of Mathematics for Modeling Human Uncer-
tainty, Springer-Verlag, Berlin, 2010.
[11] B. Liu, Uncertain risk analysis and uncertain reliablity analysis, Journal of Uncertain Sys-
tems, 4(3) (2010), 163{170.
[12] B. Liu and X. Chen, Uncertain multiobjective programming and uncertain goal programming,
Journal of Uncertainty Analysis and Applications, 3(Artical 10) (2015), 10 pages.

[13] B. Liu and K. Yao, Uncertain multilevel programming: Algorithm and applications, Com-
puters and Industrial Engineering, 89 (2015), 235{240.
[14] B. Liu, Uncertain distribution and independence of uncertain processes, Fuzzy Optimization
and Decision Making, 13(3) (2014), 259{271.
[15] B. Liu, Uncertainty Theory, 5th ed., http : ==orsc:edu:cn=liu=ut:pdf:
[16] E. J. Mcshane, Stochastic Calculus and Stochastic Models, Academic Press, New York, 1974.
[17] J. Peng, Risk metrics of loss function for uncertain system, Fuzzy Optimization and Decision
Making, 12(1) (2013), 53{64.
[18] M. Radko, Fuzzy measure and integral, Fuzzy Sets and Systems, 156(3) (2005), 365{370.
[19] A. V. Skorokhod, On a generalization of a stochastic integral, Theory of Probability & Its
Applications, 20(2) (1976), 219{233.
[20] M. Sugeno, Theorem of Fuzzy Integrals and Its Applications, Ph. D. Dissertation, Institute
of Technology, Tokyo, 1974.
[21] R. R. Yager, A foundation for a theory of possibility, Journal of Cybernetics, 10 (1980),
[22] K. Yao and X. Chen, A numerical method for solving uncertain differential equations, Journal
of Intelligent and Fuzzy Systems, 25(3) (2013), 825{832.
[23] C. You, On the convergence of uncertain sequences, Mathematical and Computer Modelling,
49(3) (2009), 482{487.
[24] C. You and L. Yan, Relationships among convergence concepts of uncertain sequences, Com-
puter Modeling and New Technologies, 20(3) (2016), 12{16.
[25] C. You and W. Wang, Some properties of complex fuzzy integral, Mathematical Problems in
Engineering, 2015(Artical ID 290539) (2015), 7 pages.
[26] C. You, H. Ma and H. Huo, A new kind of generalized fuzzy integrals, Journal of Nonlinear
Science and Applications, 9(3) (2016), 1396{1401.
[27] C. You and L. Yan, The p-distance of uncertain variables, Journal of Intelligent and Fuzzy
Systems, 32(1) (2017), 999{1006.
[28] L. A. Zadeh, Fuzzy sets, Information and Control, 8 (1965), 338{353.
[29] L. A. Zadeh, Fuzzy sets as a basis for a theory of possibility, Fuzzy Sets and Systems, 1
(1978), 3{28.