On a discrete fractional stochastic Grönwall inequality and its application in the numerical analysis of stochastic FDEs involving a martingale
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Ahmed S. Hendy
,
Mahmoud A. Zaky
and
Eid H. Doha
Abstract
The aim of this paper is to derive a novel discrete form of stochastic fractional Grönwall lemma involving a martingale. The proof of the derived inequality is accomplished by a corresponding no randomness form of the discrete fractional Grönwall inequality and an upper bound for discrete-time martingales representing the supremum in terms of the infimum. The release of a martingale term on the right-hand side of the given inequality and the graded L1 difference formula for the time Caputo fractional derivative of order 0 < α < 1 on the left-hand side are the main challenges of the stated and proved main theorem. As an example of application, the constructed theorem is used to derive an a priori estimate for a discrete stochastic fractional model at the end of the paper.
Funding source: Nazarbayev University
Award Identifier / Grant number: 091019CRP2120
Acknowledgments
The second author was supported by the Nazarbayev University Program 091019CRP2120. The second author would also like to acknowledge the partial support of the Science Committee of the Ministry of Education and Science of the Republic of Kazakhstan (Grant “Dynamical Analysis and Synchronization of Complex Neural Networks with Its Applications”).
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Author contribution: All the authors have accepted responsibility for the entire content of this submitted manuscript and approved submission.
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Research funding: This study was financially supported by Nazarbayev University Program 091019CRP2120 and the Science Committee of the Ministry of Education and Science of the Republic of Kazakhstan (Grant “Dynamical Analysis and Synchronization of Complex Neural Networks with Its Applications”).
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Conflict of interest statement: The authors declare no conflicts of interest regarding this article.
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- Frontmatter
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- A new self adaptive Tseng’s extragradient method with double-projection for solving pseudomonotone variational inequality problems in Hilbert spaces
- Solitary waves of the RLW equation via least squares method
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- Existence and uniqueness of positive solutions for fractional relaxation equation in terms of ψ-Caputo fractional derivative
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