Abstract
Canonical process is an uncertain process with stationary and independent normal increments, and the uncertain differential equation is a differential equation driven by canonical process. So far, the concept of stability in measure for uncertain differential equations has been proposed. This paper presents a concept of stability in mean for uncertain differential equations, and it gives a sufficient condition for an uncertain differential equation being stable in mean. In addition, it discusses the relationship between stability in mean and stability in measure.
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References
Bhattacharyya, R., Chatterjee, A., & Kar, S. (2010). Uncertainty theory based novel multi-objective optimization technique using embedding theorem with application to R&D project portfolio selection. Applied Mathematics, 1, 189–199.
Black, F., & Scholes, M. (1973). The pricing of options and corporate liabilities. Journal of Political Economy, 81, 637–654.
Chen, X., & Liu, B. (2010). Existence and uniqueness theorem for uncertain differential equations. Fuzzy Optimization and Decision Making, 9(1), 69–81.
Chen, X. (2011). American option pricing formula for uncertain financial market. International Journal of Operations Research, 8(2), 32–37.
Gao, X., Gao, Y., & Ralescu, D. A. (2010). On Liu’s inference rule for uncertain systems. International Journal of Uncertainty, Fuzziness and Knowledge-Based Systems, 18(1), 1–11.
Gao, Y. (2012). Existence and uniqueness theorem on uncertain differential equations with local Lipschitz condition. Journal of Uncertain Systems, 6(3), 223–232.
Ito, K. (1944). Stochastic integral (pp. 519–524). Tokyo, Japan: Proceedings of the Japan Academy.
Kalman, R. E., & Bucy, R. S. (1961). New results in linear filtering and prediction theory. Journal of Basic Engineering, 83, 95–108.
Kahneman, D., & Tversky, A. (1979). Prospect theory: An analysis of decision under risk. Econometrica, 47(2), 263–292.
Liu, B. (2007). Uncertainty theory (2nd ed.). Berlin: Springer.
Liu, B. (2008). Fuzzy process, hybrid process and uncertain process. Journal of Uncertain Systems, 2(1), 3–16.
Liu, B. (2009). Theory and Practice of Uncertain Programming (2nd ed.). Berlin: Springer.
Liu, B. (2009). Some research problems in uncertainty theory. Journal of Uncertain Systems, 3(1), 3–10.
Liu, B. (2010). Uncertainty theory: A branch of mathematics for modeling human uncertainty. Berlin: Springer.
Liu, B. (2010). Uncertain set theory and uncertain inference rule with application to uncertain control. Journal of Uncertain Systems, 4(2), 83–98.
Liu, B. (2010). Uncertain risk analysis and uncertain reliability analysis. Journal of Uncertain Systems, 4(3), 163–170.
Liu, B. (2011). Uncertain logic for modeling human language. Journal of Uncertain Systems, 5(1), 3–20.
Liu, B., & Yao, K. (2012). Uncertain integral with respect to multiple canonical processes. Journal of Uncertain Systems, 6(4), 249–254.
Liu, Y. (2012). An analytic method for solving uncertain differential equations. Journal of Uncertain Systems, 6(4), 243–248.
Peng, J., & Yao, K. (2011). A new option pricing model for stocks in uncertainty markets. International Journal of Operations Research, 8(2), 18–26.
Yao, K. (2010). Expected value of lognormal uncertain variable, Proceedings of the First International Conference on Uncertainty Theory, Urumchi, China, August 11–19, pp. 241–243.
Yao, K. (2012). Uncertain calculus with renewal process. Fuzzy Optimization and Decision Making, 11(3), 285–297.
Yao, K., Gao, J., & Gao, Y. (2013). Some stability theorems of uncertain differential equation. Fuzzy Optimization and Decision Making, 12(1), 3–13.
Yao, K. (2013). A type of uncertain differential equations with analytic solution. Journal of Uncertainty Analysis and Applications, 1(1), 1–10.
Yao, K., & Chen, X. (2013). A numerical method for solving uncertain differential equations. Journal of Intelligent & Fuzzy Systems, 25(3), 825–832.
Wiener, N. (1923). Differential space. Journal of Mathematical Physics, 2, 131–174.
Zhu, Y. (2010). Uncertain optimal control with application to a portfolio selection model. Cybernetics and Systems, 41(7), 535–547.
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This work was supported by National Natural Science Foundation of China (Grant No. 61403360, Grant No. 71371141 and Grant No. 71001080).
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Yao, K., Ke, H. & Sheng, Y. Stability in mean for uncertain differential equation. Fuzzy Optim Decis Making 14, 365–379 (2015). https://doi.org/10.1007/s10700-014-9204-2
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DOI: https://doi.org/10.1007/s10700-014-9204-2