A numerical method for solving uncertain differential equations
Citations
234 citations
161 citations
Cites background or methods from "A numerical method for solving unce..."
...Yao and Chen [23] introduced the 99-method which is a numerical method for uncertain differential equations....
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...(Yao and Chen [23]) Let α be a number with 0 < α < 1....
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...(Yao-Chen formula, Yao and Chen [23]) Let Xt and Xαt be the solution and α-path of the uncertain differential equation dXt = f (t,Xt)dt + g(t,Xt)dCt respectively....
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...(Yao-Chen formula, Yao and Chen [23]) Let Xt and X t be the solution and α-path of the uncertain differential equation dXt = f (t,Xt)dt + g(t,Xt)dCt respectively....
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...(Yao and Chen [23]) Let α be a number with 0 α 1....
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139 citations
Cites background from "A numerical method for solving unce..."
...Yao and Chen [48] proved that the solution of an uncertain differential equation can be represented by a spectrum of ordinary differential equations....
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...[39] B. S. Chen, C. S. Tseng, and H. J. Uang, “Fuzzy differential games for nonlinear stochastic systems: suboptimal approach,” IEEE Trans....
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...[47] X. Chen and B. Liu, “Existence and uniqueness theorem for uncertain differential equations,” Fuzzy Optim....
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...[38] C. Tan, Z. Jiang, X. Chen, and W. H. Ip, “A Banzhaf function for a fuzzy game,” IEEE Trans....
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...[48] K. Yao and X. Chen, “A numerical method for solving uncertain differential equations,” J. Intell....
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124 citations
Cites background or methods from "A numerical method for solving unce..."
...Then Yao and Chen [17] proposed a numerical method for solving uncertain differential equation....
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...(Yao and Chen [17]) Let α be a number with 0 < α < 1....
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...Yao and Chen [17] proposed a concept of α-path, and found a connection between an uncertain differential equation and a spectrum of ordinary differential equations....
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120 citations
References
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1,450 citations
"A numerical method for solving unce..." refers background or methods in this paper
...(Liu [5]) The uncertainty distribution of an uncertain variable ξ is defined by (x) = M{ξ ≤ x}...
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...(Liu [5]) An uncertain variable is a function from an uncertainty space ( ,L,M) to the set of real numbers, i....
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...(Liu [5]) The expected value of an uncertain variable ξ is defined by...
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...Although stochastic process, stochastic calculus and stochastic differential equation find many applications in daily life, the evolution of some undetermined phenomena behaves not like randomness but like uncertainty, which was first pointed out by Liu [5]....
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...Based on the axioms of uncertain measure, an uncertainty theory was founded by Liu [5] in 2007 and refined by Liu [8] in 2010....
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1,004 citations
"A numerical method for solving unce..." refers background or methods in this paper
...(Liu [8]) Let ξ be an uncertain variable with an uncertainty distribution ....
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...In order to calculate the expected value via inverse uncertainty distribution, Liu [8] gave the following theorem....
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...Based on the axioms of uncertain measure, an uncertainty theory was founded by Liu [5] in 2007 and refined by Liu [8] in 2010....
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987 citations
"A numerical method for solving unce..." refers background or methods in this paper
...Liu [7] verified the fundamental theorem of uncertain calculus, i....
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...Uncertain calculus, proposed by Liu [7] in 2009, is a branch of mathematics that deals with differentiation and integration of functions of uncertain processes....
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...(Liu [7]) Let Xt be an uncertain process and Ct be a canonical process....
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...Liu [7] presented a stock model in which the bond price Xt and the stock price Yt are determined by { dXt = rXtdt dYt = eYtdt + σYtdCt (10)...
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...Besides, the product uncertain measure on the product σ-algebra L was defined by Liu [7] as follows, Axiom 4: (Product Axiom) Let ( k,Lk,Mk) be uncertainty spaces for k = 1, 2, · · · Then the product uncertain measure M is an uncertain measure satisfying...
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606 citations
493 citations
"A numerical method for solving unce..." refers methods in this paper
...The numerical methods based upon this strategy can be classified as Euler-Maruyama approximation (Maruyama [12]), Milstein approximation (Milstein [13]) and Runge-Kutta approximation...
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