Fuzzy and Set-Valued Stochastic Differential Equations With Local Lipschitz Condition

2015 ◽  
Vol 23 (5) ◽  
pp. 1891-1898 ◽  
Author(s):  
Marek T. Malinowski
Keyword(s):  
Differential Equations ◽  
Stochastic Differential Equations ◽  
Lipschitz Condition ◽  
Local Lipschitz Condition

Numerical Method of Hybrid Stochastic Functional Differential Equations with the Local Lipschitz Coefficients

2011 ◽  
Vol 267 ◽  
pp. 422-426
Author(s):  
Hua Yang ◽  
Feng Jiang ◽  
Jun Hao Hu
Keyword(s):  
Differential Equations ◽  
Stochastic Differential Equations ◽  
Numerical Method ◽  
Lipschitz Condition ◽  
Numerical Solutions ◽  
Functional Differential Equations ◽  
Stochastic Functional Differential Equations ◽  
Local Lipschitz Condition ◽  
Functional Differential ◽  
Stochastic Functional

Recently, hybrid stochastic differential equations have received a great deal of attention. It is surprising that there are not any numerical schemes established for the hybrid stochastic functional differential equations. In this paper, the Euler—Maruyama method is developed, and the main aim is to show that the numerical solutions will converge to the true solutions under the local Lipschitz condition. The result obtained generalizes the earlier results.

scholarly journals Bipartite Fuzzy Stochastic Differential Equations with Global Lipschitz Condition

2016 ◽  
Vol 2016 ◽  
pp. 1-13 ◽  
Author(s):  
Marek T. Malinowski
Keyword(s):  
Differential Equations ◽  
Stochastic Differential Equations ◽  
Unique Solution ◽  
Lipschitz Condition ◽  
New Type ◽  
And Diffusion

We introduce and analyze a new type of fuzzy stochastic differential equations. We consider equations with drift and diffusion terms occurring at both sides of equations. Therefore we call them the bipartite fuzzy stochastic differential equations. Under the Lipschitz and boundedness conditions imposed on drifts and diffusions coefficients we prove existence of a unique solution. Then, insensitivity of the solution under small changes of data of equation is examined. Finally, we mention that all results can be repeated for solutions to bipartite set-valued stochastic differential equations.

scholarly journals Lp Solutions of BSDEs with Stochastic Lipschitz Condition

2007 ◽  
Vol 2007 ◽  
pp. 1-14 ◽  
Author(s):  
Jiajie Wang ◽  
Qikang Ran ◽  
Qihong Chen
Keyword(s):  
Brownian Motion ◽  
Differential Equations ◽  
Stochastic Differential Equations ◽  
Lipschitz Condition ◽  
Existence And Uniqueness ◽  
Special Class ◽  
Lipschitz Continuity ◽  
Backward Stochastic Differential Equations ◽  
Uniqueness Of The Solution ◽  
Lp Solutions

We are concerned with the solutions of a special class of backward stochastic differential equations which are driven by a Brownian motion, where the uniform Lipschitz continuity is replaced by a stochastic one. We prove the existence and uniqueness of the solution in Lp with p>1.

scholarly journals Generalized Mean-Field Fractional BSDEs With Non-Lipschitz Coefficients

2021 ◽  
Vol 10 (3) ◽  
pp. 77
Author(s):  
Qun Shi
Keyword(s):  
Brownian Motion ◽  
Differential Equations ◽  
Stochastic Differential Equations ◽  
Fractional Brownian Motion ◽  
Comparison Theorem ◽  
Lipschitz Condition ◽  
Linear Growth ◽  
Mean Field ◽  
One Dimensional ◽  
Generalized Mean

In this paper we consider one dimensional generalized mean-field backward stochastic differential equations (BSDEs) driven by fractional Brownian motion, i.e., the generators of our mean-field FBSDEs depend not only on the solution but also on the law of the solution. We first give a totally new comparison theorem for such type of BSDEs under Lipschitz condition. Furthermore, we study the existence of the solution of such mean-field FBSDEs when the coefficients are only continuous and with a linear growth.

scholarly journals Backward Doubly Stochastic Differential Equations with Markov Chains and a Comparison Theorem

Symmetry ◽  
2020 ◽  
Vol 12 (12) ◽  
pp. 1953
Author(s):  
Ning Ma ◽  
Zhen Wu
Keyword(s):  
Differential Equations ◽  
Markov Chains ◽  
Stochastic Differential Equations ◽  
Comparison Theorem ◽  
Lipschitz Condition ◽  
Existence And Uniqueness ◽  
Comparison Theorems ◽  
Yosida Approximation ◽  
Uniqueness Of Solutions ◽  
Doubly Stochastic

In this paper we study the existence and uniqueness of solutions for one kind of backward doubly stochastic differential equations (BDSDEs) with Markov chains. By generalizing the Itô’s formula, we study such problem under the Lipschitz condition. Moreover, thanks to the Yosida approximation, we solve such problem under monotone condition. Finally, we give the comparison theorems for such equations under the above two conditions respectively.

scholarly journals Ulam–Hyers Stability of Caputo-Type Fractional Stochastic Differential Equations with Time Delays

2021 ◽  
Vol 2021 ◽  
pp. 1-24
Author(s):  
Xue Wang ◽  
Danfeng Luo ◽  
Zhiguo Luo ◽  
Akbar Zada
Keyword(s):  
Differential Equations ◽  
Stochastic Differential Equations ◽  
Lipschitz Condition ◽  
Existence And Uniqueness ◽  
Time Delays ◽  
Jensen’S Inequality ◽  
Uniqueness Of Solutions ◽  
Gronwall’S Inequality ◽  
Fractional Stochastic Differential Equations ◽  
New Criteria

In this paper, we study a class of Caputo-type fractional stochastic differential equations (FSDEs) with time delays. Under some new criteria, we get the existence and uniqueness of solutions to FSDEs by Carath e ´ odory approximation. Furthermore, with the help of H o ¨ lder’s inequality, Jensen’s inequality, It o ^ isometry, and Gronwall’s inequality, the Ulam–Hyers stability of the considered system is investigated by using Lipschitz condition and non-Lipschitz condition, respectively. As an application, we give two representative examples to show the validity of our theories.

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