scholarly journals On a Coupled System of Stochastic Ito^-Differential and the Arbitrary (Fractional) Order Differential Equations with Nonlocal Random and Stochastic Integral Conditions

Mathematics ◽  
2021 ◽  
Vol 9 (20) ◽  
pp. 2571
Author(s):  
A. M. A. El-Sayed ◽  
Hoda A. Fouad
Keyword(s):  
Differential Equations ◽  
Stochastic Differential Equations ◽  
Stochastic Integral ◽  
Nonlocal Conditions ◽  
Sufficient Conditions ◽  
Coupled System ◽  
Nonlocal Problems ◽  
Integral Conditions ◽  
Fractional Stochastic Differential Equations ◽  
Random Integral

The fractional stochastic differential equations had many applications in interpreting many events and phenomena of life, and the nonlocal conditions describe numerous problems in physics and finance. Here, we are concerned with the combination between the three senses of derivatives, the stochastic Ito^-differential and the fractional and integer orders derivative for the second order stochastic process in two nonlocal problems of a coupled system of two random and stochastic differential equations with two nonlocal stochastic and random integral conditions and a coupled system of two stochastic and random integral conditions. We study the existence of mean square continuous solutions of these two nonlocal problems by using the Schauder fixed point theorem. We discuss the sufficient conditions and the continuous dependence for the unique solution.

scholarly journals On a Coupled System of Random and Stochastic Differential Equations with Nonlocal Stochastic Integral Conditions

2021 ◽  
Author(s):  
A. M. A. El-Sayed ◽  
Hoda A. Foued
Keyword(s):  
Differential Equations ◽  
Continuous Dependence ◽  
Existence Of Solutions ◽  
Stochastic Integral ◽  
Nonlinear Differential Equations ◽  
Nonlocal Conditions ◽  
Coupled System ◽  
Coupled Systems ◽  
Integral Conditions ◽  
Uniqueness Of The Solution

Here we are concerning with two problems of a coupled system of random and stochastic nonlinear differential equations with two coupled systems of nonlinear nonlocal random and stochastic integral conditions. The existence of solutions will be studied. The sufficient condition for the uniqueness of the solution will be given. The continuous dependence of the unique solution on the nonlocal conditions will be proved.

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 On a Coupled System of Random and Stochastic Nonlinear Differential Equations with Coupled Nonlocal Random and Stochastic Nonlinear Integral Conditions

Mathematics ◽  
2021 ◽  
Vol 9 (17) ◽  
pp. 2111
Author(s):  
Ahmed M. A. El-Sayed ◽  
Hoda A. Fouad
Keyword(s):  
Differential Equations ◽  
Existence Of Solutions ◽  
Nonlinear Differential Equations ◽  
Nonlocal Conditions ◽  
Integral Condition ◽  
Coupled System ◽  
Coupled Systems ◽  
Integral Conditions ◽  
Uniqueness Of The Solution ◽  
Nonlocal Integral Condition

It is well known that Stochastic equations had many useful applications in describing numerous events and problems of real world, and the nonlocal integral condition is important in physics, finance and engineering. Here we are concerned with two problems of a coupled system of random and stochastic nonlinear differential equations with two coupled systems of nonlinear nonlocal random and stochastic integral conditions. The existence of solutions will be studied. The sufficient condition for the uniqueness of the solution will be given. The continuous dependence of the unique solution on the nonlocal conditions will be proved.

scholarly journals Fuzzy stochastic differential equations driven by fractional Brownian motion

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Hossein Jafari ◽  
Marek T. Malinowski ◽  
M. J. Ebadi
Keyword(s):  
Brownian Motion ◽  
Differential Equations ◽  
Stochastic Differential Equations ◽  
Fractional Brownian Motion ◽  
Approximation Method ◽  
Existence And Uniqueness ◽  
Financial Models ◽  
Stochastic Integral ◽  
Long Range Dependence ◽  
World Systems

AbstractIn this paper, we consider fuzzy stochastic differential equations (FSDEs) driven by fractional Brownian motion (fBm). These equations can be applied in hybrid real-world systems, including randomness, fuzziness and long-range dependence. Under some assumptions on the coefficients, we follow an approximation method to the fractional stochastic integral to study the existence and uniqueness of the solutions. As an example, in financial models, we obtain the solution for an equation with linear coefficients.

scholarly journals Conditions to Guarantee the Existence of the Solution to Stochastic Differential Equations of Neutral Type

Symmetry ◽  
2020 ◽  
Vol 12 (10) ◽  
pp. 1613
Author(s):  
Mun-Jin Bae ◽  
Chan-Ho Park ◽  
Young-Ho Kim
Keyword(s):  
Differential Equations ◽  
Stochastic Differential Equations ◽  
Existence And Uniqueness ◽  
Neutral Type ◽  
Sufficient Conditions ◽  
Exponential Estimates ◽  
Analysis Theory ◽  
Linear Growth Condition ◽  
Uniqueness Of The Solution ◽  
The Uniqueness Theorem

The main purpose of this study was to demonstrate the existence and the uniqueness theorem of the solution of the neutral stochastic differential equations under sufficient conditions. As an alternative to the stochastic analysis theory of the neutral stochastic differential equations, we impose a weakened Ho¨lder condition and a weakened linear growth condition. Stochastic results are obtained for the theory of the existence and uniqueness of the solution. We first show that the conditions guarantee the existence and uniqueness; then, we show some exponential estimates for the solutions.

scholarly journals Existence and Stability of Solutions of Fuzzy Fractional Stochastic Differential Equations with Fractional Brownian Motions

2021 ◽  
Vol 2021 ◽  
pp. 1-9
Author(s):  
Elhoussain Arhrrabi ◽  
M’hamed Elomari ◽  
Said Melliani ◽  
Lalla Saadia Chadli
Keyword(s):  
Brownian Motion ◽  
Differential Equations ◽  
Exponential Stability ◽  
Stochastic Differential Equations ◽  
Fractional Brownian Motion ◽  
Stability Of Solutions ◽  
Brownian Motions ◽  
Fractional Brownian Motions ◽  
Existence And Stability ◽  
Fractional Stochastic Differential Equations

The existence, uniqueness, and stability of solutions to fuzzy fractional stochastic differential equations (FFSDEs) driven by a fractional Brownian motion (fBm) with the Lipschitzian condition are investigated. Finally, we investigate the exponential stability of solutions.

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