Abstract
Multifactor uncertain differential equation driven by multiple Liu processes is a type of uncertain differential equations. In this paper, a concept of stability in distribution for multifactor uncertain differential equation is introduced, and a sufficient condition for a multifactor uncertain differential equation being stable in distribution is proved. The relationships among stability in distribution, stability in measure, and stability in mean for multifactor uncertain differential equation are also discussed. Moreover, some examples are given to illustrate the stability in distribution.
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Chen X, Gao J (2013) Uncertain term structure model of interest rate. Soft Comput 17(4):597–604
Chen X, Liu B (2010) Existence and uniqueness theorem for uncertain differential equations. Fuzzy Optim Decis Mak 9(1):69–81
Chen X, Ralescu DA (2012) B-spline method of uncertain statistics with applications to estimate travel distance. J Uncertain Syst 6(4):256–262
Chen X, Ralescu DA (2013) Liu process and uncertain calculus. Uncertainty Anal Appl 1:3
Gao J, Yang X, Liu D (2016) Uncertain Shapley value of coalitional game with application to supply chain alliance. Appl Soft Comput. doi:10.1016/j.asoc.2016.06.018
Gao R (2017) Uncertain wave equation with infinite half-boundary. Appl Math Comput 304:28–40
Gao R (2016) Milne method for solving uncertain differential equations. Appl Math Comput 274(11):774–785
Gao R, Chen X (2017) Some concepts and properties of uncertain fields. J Intell Fuzzy Syst. doi:10.3233/JIFS-16314
Gao R, Sheng Y (2016) Law of large numbers for uncertain random variables with different chance distributions. J Intell Fuzzy Syst 31(3):1227–1234
Gao R, Sun Y, Ralescu D (2016) Order statistics of uncertain random variables with application to k-out-of-n system. Fuzzy Optim Decis Mak. doi:10.1007/s10700-016-9245-9
Gao R, Yao K (2016) Importance index of components in uncertain random systems. Knowl Based Syst 109:208–217
Gao Y, Yang X, Fu Z (2017) Lookback option pricing problem of uncertain exponential Ornstein–Uhlenbeck model. Soft Comput. doi:10.1007/s00500-017-2558-y
Ke H, Huang H, Gao X (2017) Pricing decision problem in dual-channel supply chain based on experts’ belief degrees. Soft Comput. doi:10.1007/s00500-017-2600-0
Li S, Peng J, Zhang B (2015) Multifactor uncertain differential equation. J Uncertainty Anal Appl 3(7):1–19
Liu B (2007) Uncertainty theory, 2nd edn. Springer, Berlin
Liu B (2009) Some research problems in uncertainty theory. J Uncertain Syst 3(1):3–10
Liu B (2015) Uncertainty theory, 4th edn. Springer, Berlin
Liu B (2012) Why is there a need for uncertainty theory. J Uncertain Syst 6(1):3–10
Liu B (2008) Fuzzy process, hybrid process and uncertain process. J Uncertain Syst 2(1):3–16
Liu B (2014) Uncertainty distribution and independence on uncertain processes. Fuzzy Optim Decis Mak 13(3):259–271
Liu B (2010) Uncertainty theory: a branch of mathematics for modeling human uncertainty. Springer, Berlin
Liu B, Yao K (2012) Uncertain integral with respect to multiple canonical process. J Uncertain Syst 6(4):249–254
Liu Y, Chen X, Ralescu D (2015) Uncertain currency model and currency option pricing. Int J Intell Syst 30(1):40–51
Liu H, Ke H, Fei W (2014) Almost sure stability for uncertain differential equation. Fuzzy Optim Decis Mak 13(4):463–473
Peng Z, Iwamura K (2010) A sufficient and necessary condition of uncertainty distribution. J Interdiscip Math 13(3):277–285
Sheng Y, Gao J (2016) Exponential stability of uncertain differential equation. Soft Comput 20:3673–3678
Sheng Y, Shi G, Cui Q (2017) Almost sure stability for multifactor uncertain differential equation. J Intell Fuzzy Syst 32:2187–2194
Sheng Y, Wang C (2014) Stability in \(p\)-th moment for uncertain differential equation. J Intell Fuzzy Syst 26(3):1263–1271
Yao K (2013) Extreme values and integral of solution of uncertain differential equation. J Uncertainty Anal Appl 1:2
Yao K (2015) Uncertain contour process and its application in stock model with floating interest rate. Fuzzy Optim Decis Mak 14(4):399–424
Yao K, Chen X (2013) A numerical method for solving uncertain differential equations. J Intell Fuzzy Syst 25(3):825–832
Yao K, Gao J, Gao Y (2013) Some stability theorems of uncertain differential equation. Fuzzy Optim Decis Mak 12(1):3–13
Yao K, Ke H, Sheng Y (2015) Stability in mean for uncertain differential equation. Fuzzy Optim Decis Mak 14(3):365–379
Yao K (2016) Uncertain differential equations. Springer, Berlin
Yao K, Liu B (2017) Uncertain regression analysis: an approach for imprecise observations. Soft Comput. doi:10.1007/s00500-017-2521-y
Yang X, Gao J (2013) Uncertain differential games with application to capitalism. J Uncertainty Anal Appl 1:17
Yang X, Gao J (2016) Linear-quadratic uncertain differential games with application to resource extraction problem. IEEE Trans Fuzzy Syst 24(4):819–826
Yang X, Gao J (2017) Bayesian equilibria for uncertain bimatrix game with asymmetric information. J Intell Manuf 28(3):515–525
Yang X, Ni Y, Zhang Y (2017) Stability in inverse distribution for uncertain differential equtions. J Intell Fuzzy Syst 32:2051–2059
Yang X, Ralescu D (2015) Adams method for solving uncertain differential equations. Appl Math Comput 270:993–1003
Yang X, Shen Y (2015) Runge–Kutta method for solving uncertain differential equations. J Uncertainty Anal Appl 3:17
Yang X, Yao K (2016) Uncertain partial differential equation with application to heat conduction. Fuzzy Optim Decis Mak. doi:10.1007/s10700-016-9253-9
Zhang Z, Gao R, Yang X (2016) The stability of multifactor uncertain differential equation. J Intell Fuzzy Syst 30:3281–3290
Zhu Y (2010) Uncertain optimal control with application to a portfolio selection model. Cybern Syst 41(7):535–547
Acknowledgements
This work was supported by the Natural Science Foundation of China (No. 11626234), the Higher Educational Science and Technology Program Foundation of Shandong Province (No. J13LI10), and the Foundation of Liaocheng University (No. 318011303).
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Ma, W., Liu, L., Gao, R. et al. Stability in distribution for multifactor uncertain differential equation. J Ambient Intell Human Comput 8, 707–716 (2017). https://doi.org/10.1007/s12652-017-0517-1
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DOI: https://doi.org/10.1007/s12652-017-0517-1