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Stability in mean for uncertain differential equation. (English) Zbl 1463.93250
Summary: 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.

MSC:
93E15 Stochastic stability in control theory
93C41 Control/observation systems with incomplete information
34D20 Stability of solutions to ordinary differential equations
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[1] Bhattacharyya, R; Chatterjee, A; Kar, S, Uncertainty theory based novel multi-objective optimization technique using embedding theorem with application to R&D project portfolio selection, Applied Mathematics, 1, 189-199, (2010)
[2] Black, F; Scholes, M, The pricing of options and corporate liabilities, Journal of Political Economy, 81, 637-654, (1973) · Zbl 1092.91524
[3] Chen, X; Liu, B, Existence and uniqueness theorem for uncertain differential equations, Fuzzy Optimization and Decision Making, 9, 69-81, (2010) · Zbl 1196.34005
[4] Chen, X, American option pricing formula for uncertain financial market, International Journal of Operations Research, 8, 32-37, (2011)
[5] 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. · Zbl 1207.68386
[6] Gao, Y, Existence and uniqueness theorem on uncertain differential equations with local Lipschitz condition, Journal of Uncertain Systems, 6, 223-232, (2012)
[7] Ito, K. (1944). Stochastic integral (pp. 519-524). Tokyo, Japan: Proceedings of the Japan Academy.
[8] Kalman, RE; Bucy, RS, New results in linear filtering and prediction theory, Journal of Basic Engineering, 83, 95-108, (1961)
[9] Kahneman, D; Tversky, A, Prospect theory: an analysis of decision under risk, Econometrica, 47, 263-292, (1979) · Zbl 0411.90012
[10] Liu, B. (2007). Uncertainty theory (2nd ed.). Berlin: Springer.
[11] Liu, B, Fuzzy process, hybrid process and uncertain process, Journal of Uncertain Systems, 2, 3-16, (2008)
[12] Liu, B. (2009). Theory and Practice of Uncertain Programming (2nd ed.). Berlin: Springer. · Zbl 1158.90010
[13] Liu, B, Some research problems in uncertainty theory, Journal of Uncertain Systems, 3, 3-10, (2009)
[14] Liu, B. (2010). Uncertainty theory: A branch of mathematics for modeling human uncertainty. Berlin: Springer.
[15] Liu, B, Uncertain set theory and uncertain inference rule with application to uncertain control, Journal of Uncertain Systems, 4, 83-98, (2010)
[16] Liu, B, Uncertain risk analysis and uncertain reliability analysis, Journal of Uncertain Systems, 4, 163-170, (2010)
[17] Liu, B, Uncertain logic for modeling human language, Journal of Uncertain Systems, 5, 3-20, (2011)
[18] Liu, B; Yao, K, Uncertain integral with respect to multiple canonical processes, Journal of Uncertain Systems, 6, 249-254, (2012)
[19] Liu, Y, An analytic method for solving uncertain differential equations, Journal of Uncertain Systems, 6, 243-248, (2012)
[20] Peng, J; Yao, K, A new option pricing model for stocks in uncertainty markets, International Journal of Operations Research, 8, 18-26, (2011)
[21] 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.
[22] Yao, K, Uncertain calculus with renewal process, Fuzzy Optimization and Decision Making, 11, 285-297, (2012) · Zbl 1277.60144
[23] Yao, K; Gao, J; Gao, Y, Some stability theorems of uncertain differential equation, Fuzzy Optimization and Decision Making, 12, 3-13, (2013)
[24] Yao, K. (2013). A type of uncertain differential equations with analytic solution. Journal of Uncertainty Analysis and Applications, \(1\)(1), 1-10.
[25] Yao, K; Chen, X, A numerical method for solving uncertain differential equations, Journal of Intelligent & Fuzzy Systems, 25, 825-832, (2013) · Zbl 1291.65025
[26] Wiener, N, Differential space, Journal of Mathematical Physics, 2, 131-174, (1923)
[27] Zhu, Y, Uncertain optimal control with application to a portfolio selection model, Cybernetics and Systems, 41, 535-547, (2010) · Zbl 1225.93121
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