scholarly journals Carathéodory’s approximate solution to stochastic differential delay equation

Filomat ◽  
2016 ◽  
Vol 30 (7) ◽  
pp. 2019-2028
Author(s):  
Young-Ho Kim
Keyword(s):  
Approximate Solution ◽  
Growth Condition ◽  
Lipschitz Condition ◽  
Linear Growth ◽  
Approximate Solutions ◽  
Delay Equation ◽  
Differential Delay Equation ◽  
Differential Delay ◽  
Stochastic Differential Delay Equation ◽  
Linear Growth Condition

The main aim of this paper is to discuss Carath?odory?s and Euler-Maruyama?s approximate solutions to stochastic differential delay equation. To make the theory more understandable, we impose the non-uniform Lipschitz condition and non-linear growth condition.

scholarly journals Fixed Point Theorems Applied in Uncertain Fractional Differential Equation with Jump

Symmetry ◽  
2020 ◽  
Vol 12 (5) ◽  
pp. 765
Author(s):  
Zhifu Jia ◽  
Xinsheng Liu ◽  
Cunlin Li
Keyword(s):  
Differential Equation ◽  
Fixed Point ◽  
Fixed Point Theorem ◽  
Fractional Differential Equation ◽  
Growth Condition ◽  
Lipschitz Condition ◽  
Point Theorem ◽  
Linear Growth ◽  
Linear Growth Condition ◽  
Fractional Differential

No previous study has involved uncertain fractional differential equation (FDE, for short) with jump. In this paper, we propose the uncertain FDEs with jump, which is driven by both an uncertain V-jump process and an uncertain canonical process. First of all, for the one-dimensional case, we give two types of uncertain FDEs with jump that are symmetric in terms of form. The next, for the multidimensional case, when the coefficients of the equations satisfy Lipschitz condition and linear growth condition, we establish an existence and uniqueness theorems of uncertain FDEs with jump of Riemann-Liouville type by Banach fixed point theorem. A symmetric proof in terms of form is suitable to the Caputo type. When the coefficients do not satisfy the Lipschitz condition and linear growth condition, we just prove an existence theorem of the Caputo type equation by Schauder fixed point theorem. In the end, we present an application about uncertain interest rate model.

scholarly journals Stochastic fractional integro-differential equations with weakly singular kernels: Well-posedness and Euler–Maruyama approximation

2021 ◽  
Vol 0 (0) ◽  
pp. 0
Author(s):  
Xinjie Dai ◽  
Aiguo Xiao ◽  
Weiping Bu
Keyword(s):  
Differential Equations ◽  
Growth Condition ◽  
Lipschitz Condition ◽  
Linear Growth ◽  
Well Posedness ◽  
Initial Value ◽  
Weakly Singular Kernels ◽  
Weakly Singular ◽  
Linear Growth Condition ◽  
Singular Kernels

<p style='text-indent:20px;'>This paper considers the initial value problem of general nonlinear stochastic fractional integro-differential equations with weakly singular kernels. Our effort is devoted to establishing some fine estimates to include all the cases of Abel-type singular kernels. Firstly, the existence, uniqueness and continuous dependence on the initial value of the true solution under local Lipschitz condition and linear growth condition are derived in detail. Secondly, the Euler–Maruyama method is developed for solving numerically the equation, and then its strong convergence is proven under the same conditions as the well-posedness. Moreover, we obtain the accurate convergence rate of this method under global Lipschitz condition and linear growth condition. In particular, the Euler–Maruyama method can reach strong first-order superconvergence when <inline-formula><tex-math id="M1">\begin{document}$ \alpha = 1 $\end{document}</tex-math></inline-formula>. Finally, several numerical tests are reported for verification of the theoretical findings.</p>

scholarly journals Euler-Maruyama approximation of backward doubly stochastic differential delay equations

2016 ◽  
Vol 5 (3) ◽  
pp. 146
Author(s):  
Falah Sarhan ◽  
LIU JICHENG
Keyword(s):  
Numerical Solutions ◽  
Delay Equations ◽  
Delay Equation ◽  
Differential Delay Equation ◽  
True Solution ◽  
Differential Delay ◽  
Doubly Stochastic ◽  
Stochastic Differential Delay Equation ◽  
Differential Delay Equations ◽  
Stochastic Differential Delay Equations

In this paper, we attempt to introduce a new numerical approach to solve backward doubly stochastic differential delay equation ( shortly-BDSDDEs ). In the beginning, we present some assumptions to get the numerical scheme for BDSDDEs, from which we prove important theorem. We use the relationship between backward doubly stochastic differential delay equations and stochastic controls by interpreting BDSDDEs as some stochastic optimal control problems, to solve the approximated BDSDDEs and we prove that the numerical solutions of backward doubly stochastic differential delay equation converge to the true solution under the Lipschitz condition.

Sign in / Sign up

Export Citation Format

Share Document