scholarly journals (Weighted pseudo) almost automorphic solutions in distribution for fractional stochastic differential equations driven by levy noise

Filomat ◽  
2021 ◽  
Vol 35 (7) ◽  
pp. 2403-2424
Author(s):  
Min Yang
Keyword(s):  
Differential Equations ◽  
Fractional Calculus ◽  
Stochastic Differential Equations ◽  
Existence And Uniqueness ◽  
Stochastic Analysis ◽  
Contraction Principle ◽  
Lévy Noise ◽  
Almost Automorphic ◽  
Fractional Stochastic Differential Equations ◽  
Pseudo Almost Automorphic

In this paper, by using contraction principle, fractional calculus and stochastic analysis, we study the existence and uniqueness of (weighted pseudo) almost automorphic solutions in distribution for fractional stochastic differential equations driven by L?vy noise. An example is presented to illustrate the application of the abstract results.

scholarly journals Pseudo Almost Automorphic Solutions for Stochastic Differential Equations Driven by Lévy Noise and Its Optimal Control

Symmetry ◽  
2021 ◽  
Vol 13 (9) ◽  
pp. 1674
Author(s):  
Chao Tang ◽  
Rong Hou
Keyword(s):  
Optimal Control ◽  
Differential Equations ◽  
Stochastic Processes ◽  
Stochastic Differential Equations ◽  
Lévy Noise ◽  
Bounded Interval ◽  
Almost Automorphic ◽  
Pseudo Almost Automorphic ◽  
Levy Noise ◽  
Theoretical Results

As we know, the periodic functions are symmetric within a cycle time, and it is meaningful to generalize the periodicity into more general cases, such as almost periodicity or almost automorphy. In this work, we introduce the concept of Poisson Sγ2-pseudo almost automorphy (or Poisson generalized Stepanov-like pseudo almost automorphy) for stochastic processes, which are almost-symmetric within a suitable period, and establish some useful properties of such stochastic processes, including the composition theorems. In addition, we apply a Krasnoselskii–Schaefer type fixed point theorem to obtain the existence of pseudo almost automorphic solutions in distribution for some semilinear stochastic differential equations driven by Lévy noise under Sγ2-pseudo almost automorphic coefficients. In addition, then we establish optimal control results on the bounded interval. Finally, an example is provided to illustrate the theoretical results obtained in this paper.

scholarly journals Existence and Uniqueness of Mild Solutions for Nonlinear Stochastic Impulsive Differential Equation

2011 ◽  
Vol 2011 ◽  
pp. 1-10
Author(s):  
L. J. Shen ◽  
J. T. Sun
Keyword(s):  
Differential Equation ◽  
Differential Equations ◽  
Stochastic Differential Equations ◽  
Existence And Uniqueness ◽  
Stochastic Analysis ◽  
Impulsive Differential Equation ◽  
Mild Solutions ◽  
Sufficient Conditions ◽  
Impulsive Differential Equations ◽  
Analysis Technique

This paper investigates the existence and uniqueness of mild solutions to the general nonlinear stochastic impulsive differential equations. By using Schaefer's fixed theorem and stochastic analysis technique, we propose sufficient conditions on existence and uniqueness of solution for stochastic differential equations with impulses. An example is also discussed to illustrate the effectiveness of the obtained results.

scholarly journals On the Existence and Uniqueness of Solutions for Multidimensional Fractional Stochastic Differential Equations with Variable Order

Mathematics ◽  
2021 ◽  
Vol 9 (17) ◽  
pp. 2106
Author(s):  
Seyfeddine Moualkia ◽  
Yong Xu
Keyword(s):  
Differential Equations ◽  
Stochastic Differential Equations ◽  
Fractional Order ◽  
Existence And Uniqueness ◽  
Sufficient Conditions ◽  
Variable Order ◽  
Uniqueness Of Solutions ◽  
Fractional Stochastic Differential Equations ◽  
Picard Iterations

Fractional stochastic differential equations are still in their infancy. Based on some existing results, the main difficulties here are how to deal with those equations if the fractional order is varying with time and how to confirm the existence of their solutions in this case. This paper is about the existence and uniqueness of solutions to the fractional stochastic differential equations with variable order. We prove the existence by using the Picard iterations and propose new sufficient conditions for the uniqueness.

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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