Existence and uniqueness theorem for uncertain differential equations

Chen, X.; Liu, B.

doi:10.1007/s10700-010-9073-2

Cited by 364 publications

(157 citation statements)

References 7 publications

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“…This conclusion is contradictory and unacceptable to our judgment. To deal with the above situation, Liu [12][13][14][15] proposed an uncertain measure and further developed the uncertainty theory, which has been used in various areas (e.g., insurance, medical care, environment and education) especially in the study of portfolio optimization [16][17][18][19]. Qin, et al [20] first studied mean-variance model in the uncertain environment.…”

Section: Introductionmentioning

confidence: 99%

A Risk-Free Protection Index Model for Portfolio Selection with Entropy Constraint under an Uncertainty Framework

Gao

Huicheng

2017

Entropy

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This paper aims to develop a risk-free protection index model for portfolio selection based on the uncertain theory. First, the returns of risk assets are assumed as uncertain variables and subject to reputable experts' evaluations. Second, under this assumption, combining with the risk-free interest rate we define a risk-free protection index (RFPI), which can measure the protection degree when the loss of risk assets happens. Third, note that the proportion entropy serves as a complementary means to reduce the risk by the preset diversification requirement. We put forward a risk-free protection index model with an entropy constraint under an uncertainty framework by applying the RFPI, Huang's risk index model (RIM), and mean-variance-entropy model (MVEM). Furthermore, to solve our portfolio model, an algorithm is given to estimate the uncertain expected return and standard deviation of different risk assets by applying the Delphi method. Finally, an example is provided to show that the risk-free protection index model performs better than the traditional MVEM and RIM.

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Section: Introductionmentioning

confidence: 99%

A Risk-Free Protection Index Model for Portfolio Selection with Entropy Constraint under an Uncertainty Framework

Gao

Huicheng

2017

Entropy

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“…Liu [11] initiated the study of uncertain differential equations. Afterwards, Chen and Liu [12] proved the existence and uniqueness theorem for uncertain differential equations. Some stability theorems were proved by Yao et al [13].…”

Section: Introductionmentioning

confidence: 99%

Valuing Interest Rate Swap Contracts in Uncertain Financial Market

Chen

Zhang

2016

Sustainability

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Swap is a financial contract between two counterparties who agree to exchange one cash flow stream for another, according to some predetermined rules. When the cash flows are fixed rate interest and floating rate interest, the swap is called an interest rate swap. This paper investigates two valuation models of the interest rate swap contracts in the uncertain financial market. The new models are based on belief degrees, and require relatively less historical data compared to the traditional probability models. The first valuation model is designed for a mean-reversion term structure, while the second is designed for a term structure with hump effect. Explicit solutions are developed by using the Yao-Chen formula. Moreover, a numerical method is designed to calculate the value of the interest rate swap alternatively. Finally, two examples are given to show their applications and comparisons.

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“…The existence and uniqueness of an uncertain differential equation was studied by Chen and Liu [11] in 2010. An uncertain differential equation has a unique solution if its coefficients satisfy Lipschitz condition and linear growth condition.…”

Section: Introductionmentioning

confidence: 99%

“…Chen and Liu [11] figured out the analytic solution of the linear uncertain differential equation. Liu [17] and Yao [18] considered a spectrum of analytic methods to solve some special classes of nonlinear uncertain differential equations.…”

Section: Introductionmentioning

confidence: 99%

Runge-Kutta Method for Solving Uncertain Differential Equations

Yang

Shen

2015

J. Uncertain. Anal. Appl.

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Uncertain differential equations have been widely applied to many fields especially to uncertain finance. Unfortunately, we cannot always get the analytic solution of uncertain differential equations. Early researchers have put up a numerical method based on the Euler method. This paper designs a new numerical method for solving uncertain differential equations via the widely-used Runge-Kutta method. Some examples are given to illustrate the effectiveness of the Runge-Kutta method when calculating the uncertainty distribution, expected value, extreme value, and time integral of solution of uncertain differential equations.

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Existence and uniqueness theorem for uncertain differential equations

Cited by 364 publications

References 7 publications

A Risk-Free Protection Index Model for Portfolio Selection with Entropy Constraint under an Uncertainty Framework

A Risk-Free Protection Index Model for Portfolio Selection with Entropy Constraint under an Uncertainty Framework

Valuing Interest Rate Swap Contracts in Uncertain Financial Market

Runge-Kutta Method for Solving Uncertain Differential Equations

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