The existence and averaging principle for Caputo fractional stochastic delay differential systems

Li, Mengmeng; Wang, JinRong

doi:10.1007/s13540-023-00146-3

Cited by 8 publications

(7 citation statements)

References 24 publications

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“…Remark 1. If p = 2 and f ≡ 0, then FSDDEs (1) reduces to FSDDSs (1.1) in [16]. Therefore, Theorems 1 and 2 generalize the main results of [16].…”

Section: An Averaging Principlesupporting

confidence: 53%

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The Existence and Averaging Principle for Caputo Fractional Stochastic Delay Differential Systems with Poisson Jumps

Bai,

Bai

2024

Axioms

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In this paper, we obtain the existence and uniqueness theorem for solutions of Caputo-type fractional stochastic delay differential systems(FSDDSs) with Poisson jumps by utilizing the delayed perturbation of the Mittag–Leffler function. Moreover, by using the Burkholder–Davis–Gundy inequality, Doob’s martingale inequality, and Hölder inequality, we prove that the solution of the averaged FSDDSs converges to that of the standard FSDDSs in the sense of Lp. Some known results in the literature are extended.

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“…Remark 1. If p = 2 and f ≡ 0, then FSDDEs (1) reduces to FSDDSs (1.1) in [16]. Therefore, Theorems 1 and 2 generalize the main results of [16].…”

Section: An Averaging Principlesupporting

confidence: 53%

“…Recently, Li and Wang in [16] investigated the existence, uniqueness, and averaging principle for the following Caputo-type FSDDEs:…”

Section: Introductionmentioning

confidence: 99%

The Existence and Averaging Principle for Caputo Fractional Stochastic Delay Differential Systems with Poisson Jumps

Bai,

Bai

2024

Axioms

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“…However, it must be noted that the convergence rate is ε 2α−1−γ instead of ε 1−γ unless α = 1. In other words, the convergence rate is related to the fractional order; this point is ignored in the existing literature due to the misuse of the standard form of fractional stochastic differential equations [3][4][5][6][7]9,10,15,16].…”

Section: Resultsmentioning

confidence: 99%

“…Recently, Yang et al [14] developed an averaging principle for Hilfer FSDEs in the infinite-dimensional space. Li and Wang [15] presented an averaging theorem for Caputo FSDDEs driven by Gaussian white noise. Bai et al [16] proved an averaging result for Caputo FSDDEs with Poisson noise.…”

Section: Introductionmentioning

confidence: 99%

A Note on Averaging Principles for Fractional Stochastic Differential Equations

Liu,

Zhang,

Wang

et al. 2024

Fractal Fract

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Over the past few years, many scholars began to study averaging principles for fractional stochastic differential equations since they can provide an approximate analytical method to reduce such systems. However, in the most previous studies, there is a misunderstanding of the standard form of fractional stochastic differential equations, which consequently causes the wrong estimation of the convergence rate. In this note, we take fractional stochastic differential equations with Lévy noise as an example to clarify these two issues. The corrections herein have no effect on the main proofs except the two points mentioned above. The innovation of this paper lies in three aspects: (i) the standard form of the fractional stochastic differential equations is derived under natural time scale; (ii) it is first proved that the convergence interval and rate are related to the fractional order; and (iii) the presented results contain and improve some well known research achievements.

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“…Recently, Liu et al [20] gave the exact solutions of a class of fractional delay differential equations. Li and Wang in [21] studied the existence and uniqueness of a class of Caputo fractional stochastic delay differential systems (FSDDSs). In [22], we extended the main results of [21].…”

Section: Introductionmentioning

confidence: 99%

Hyers–Ulam Stability of Caputo Fractional Stochastic Delay Differential Systems with Poisson Jumps

Bai,

Bai

2024

Mathematics

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In this paper, we explore the stability of a new class of Caputo-type fractional stochastic delay differential systems with Poisson jumps. We prove the Hyers–Ulam stability of the solution by utilizing a version of fixed point theorem, fractional calculus, Cauchy–Schwartz inequality, Jensen inequality, and some stochastic analysis techniques. Finally, an example is provided to illustrate the effectiveness of the results.

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The existence and averaging principle for Caputo fractional stochastic delay differential systems

Cited by 8 publications

References 24 publications

The Existence and Averaging Principle for Caputo Fractional Stochastic Delay Differential Systems with Poisson Jumps

The Existence and Averaging Principle for Caputo Fractional Stochastic Delay Differential Systems with Poisson Jumps

A Note on Averaging Principles for Fractional Stochastic Differential Equations

Hyers–Ulam Stability of Caputo Fractional Stochastic Delay Differential Systems with Poisson Jumps

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