Abstract
Canonical process is a type of uncertain process with stationary and independent increments which are normal uncertain variables, and uncertain differential equation is a type of differential equation driven by canonical process. This paper will give a theorem on the Lipschitz continuity of canonical process based on which this paper will also provide a sufficient condition for an uncertain differential equation being stable.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Black F., Scholes M. (1973) The pricing of options and corporate liabilities. Journal of Political Economy 81: 637–654
Chen X. (2011) American option pricing formula for uncertain financial market. International Journal of Operations Research 8(2): 32–37
Chen X., Liu B. (2010) Existence and uniqueness theorem for uncertain differential equations. Fuzzy Optimization and Decision Making 9(1): 69–81
Ito, K. (1944). Stochastic integral. In Proceedings of the Japan Academy, Tokyo, Japan (pp. 519–524).
Liu B. (2007) Uncertainty theory. Springer, Berlin
Liu B. (2008) Fuzzy process, hybrid process and uncertain process. Journal of Uncertain Systems 2(1): 3–16
Liu B. (2009) Some research problems in uncertainty theory. Journal of Uncertain Systems 3(1): 3–10
Liu B. (2011) Uncertainty theory: A branch of mathematics for modeling human uncertainty. Springer, Berlin
Liu B. (2012) Why is there a need for uncertainty theory. Journal of Uncertain Systems 6(1): 3–10
Liu, B. (2012b). Uncertainty theory (4th ed). http://orsc.edu.cn/liu/ut.pdf.
Liu, B. (2012c). Toward uncertain finance theory via uncertain calculus (Technical Report).
Liu Y. H., Ha M. H. (2010) Expected value of function of uncertain variables. Journal of Uncertain Systems 4(3): 181–186
Peng J., Yao K. (2011) A new option pricing model for stocks in uncertainty markets. International Journal of Operations Research 8(2): 18–26
Wiener N. (1923) Differential space. Journal of Mathematical Physics 2: 131–174
Yao K. (2012) Uncertain calculus with renewal process. Fuzzy Optimization and Decision Making 11(3): 285–297
Zhu Y. (2010) Uncertain optimal control with application to a portfolio selection model. Cybernetics and Systems 41(7): 535–547
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Yao, K., Gao, J. & Gao, Y. Some stability theorems of uncertain differential equation. Fuzzy Optim Decis Making 12, 3–13 (2013). https://doi.org/10.1007/s10700-012-9139-4
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10700-012-9139-4