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doi: 10.3934/eect.2022012
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Impulsive conformable fractional stochastic differential equations with Poisson jumps

Higher Institute of Engineering, El-Shorouk Academy, El-Shorouk City, Cairo, Egypt

* Corresponding author: Hamdy M. Ahmed

Received  August 2021 Revised  January 2022 Early access March 2022

In this article, periodic averaging method for impulsive conformable fractional stochastic differential equations with Poisson jumps are discussed. By using stochastic analysis, fractional calculus, Doob's martingale inequality and Cauchy-Schwarz inequality, we show that the solution of the conformable fractional impulsive stochastic differential equations with Poisson jumps converges to the corresponding averaged conformable fractional stochastic differential equations with Poisson jumps and without impulses.

Citation: Hamdy M. Ahmed. Impulsive conformable fractional stochastic differential equations with Poisson jumps. Evolution Equations and Control Theory, doi: 10.3934/eect.2022012
References:
[1]

R. AlmeidaN. R. BastosM. Teresa and T. Monteiro, Modeling some real phenomena by fractional differential equations, Mathematical Methods in the Applied Sciences, 39 (2016), 4846-4855.  doi: 10.1002/mma.3818.

[2]

S. QureshiA. YusufA. A. ShaikhM. Inc and D. Baleanu, Fractional modeling of blood ethanol concentration system with real data application, An Interdisciplinary Journal of Nonlinear Science, 29 (2019), 013143.  doi: 10.1063/1.5082907.

[3]

D. BaleanuH. Mohammadi and S. Rezapour, A fractional differential equation model for the COVID-19 transmission by using the Caputo-Fabrizio derivative, Advances in difference equations, 2020 (2020), 1-27. 

[4]

E. Bas and R. Ozarslan, Real world applications of fractional models by Atangana-Baleanu fractional derivative, IMA Chaos, Solitons and Fractals, 116 (2018), 121-125.  doi: 10.1016/j.chaos.2018.09.019.

[5]

R. KhalilM. A. HoraniA. Yousef and M. Sababheh, A new definition of fractional derivative, Journal of Computational and Applied Mathematics, 264 (2014), 65-70.  doi: 10.1016/j.cam.2014.01.002.

[6]

M. HanY. Xu and B. Pei, A mixed stochastic differential equations: Averaging principle result, Applied Mathematics Letters, 112 (2021), 106705.  doi: 10.1016/j.aml.2020.106705.

[7]

B. PeiY. Xu and J. L. Wu, Stochastic averaging for stochastic differential equations driven by fractional Brownian motion and standard Brownian motion, Applied Mathematics Letters, 100 (2020), 106006.  doi: 10.1016/j.aml.2019.106006.

[8]

J. Cui and N. Bi, Averaging principle for neutral stochastic functional differential equations with impulses and non-Lipschitz coefficients, Statistics and Probability Letters, 163 (2020), 108775.  doi: 10.1016/j.spl.2020.108775.

[9]

J. K. LiuW. Xu and Q. Guo, Averaging principle for impulsive stochastic partial differential equations, Stoch. Dynam., 21 (2021), 2150014.  doi: 10.1142/S0219493721500143.

[10]

W. Mao and X. Mao, An averaging principle for neutral stochastic functional differential equations driven by Poisson random measure, Adv. Difference Equ., 2016 (2016), 77 pp. doi: 10.1088/1751-8113/44/33/335201.

[11]

H. M. Ahmed and Q. Zhu, The averaging principle of Hilfer fractional stochastic delay differential equations with Poisson jumps, Applied Mathematics Letters, 112 (2021), 106755.  doi: 10.1016/j.aml.2020.106755.

[12]

J. Liu and W. Xu, An averaging result for impulsive fractional neutral stochastic differential equations, Applied Mathematics Letters, 114 (2021), 106892.  doi: 10.1016/j.aml.2020.106892.

[13]

W. J. XuW. Xu and S. Zhang, The averaging principle for stochastic differential equations with Caputo fractional derivative, Applied Mathematics Letters, 93 (2019), 79-84.  doi: 10.1016/j.aml.2019.02.005.

[14]

D. F. LuoQ. X. Zhu and Z. G. Luo, An averaging principle for stochastic fractional differential equations with time delays, Applied Mathematics Letters, 105 (2020), 106290.  doi: 10.1016/j.aml.2020.106290.

[15]

M. HannabouK. Hilal and A. Kajouni, Existence and uniqueness of mild solutions to impulsive nonlocal Cauchy problems, Journal of Mathematics, 2020 (2020), 5729128. 

[16]

L. Shen and J. T. Sun, Global existence of solutions for stochastic impulsive differential equations, Acta Mathematica Sinica, 27 (2011), 773-780.  doi: 10.1007/s10114-011-8650-9.

show all references

References:
[1]

R. AlmeidaN. R. BastosM. Teresa and T. Monteiro, Modeling some real phenomena by fractional differential equations, Mathematical Methods in the Applied Sciences, 39 (2016), 4846-4855.  doi: 10.1002/mma.3818.

[2]

S. QureshiA. YusufA. A. ShaikhM. Inc and D. Baleanu, Fractional modeling of blood ethanol concentration system with real data application, An Interdisciplinary Journal of Nonlinear Science, 29 (2019), 013143.  doi: 10.1063/1.5082907.

[3]

D. BaleanuH. Mohammadi and S. Rezapour, A fractional differential equation model for the COVID-19 transmission by using the Caputo-Fabrizio derivative, Advances in difference equations, 2020 (2020), 1-27. 

[4]

E. Bas and R. Ozarslan, Real world applications of fractional models by Atangana-Baleanu fractional derivative, IMA Chaos, Solitons and Fractals, 116 (2018), 121-125.  doi: 10.1016/j.chaos.2018.09.019.

[5]

R. KhalilM. A. HoraniA. Yousef and M. Sababheh, A new definition of fractional derivative, Journal of Computational and Applied Mathematics, 264 (2014), 65-70.  doi: 10.1016/j.cam.2014.01.002.

[6]

M. HanY. Xu and B. Pei, A mixed stochastic differential equations: Averaging principle result, Applied Mathematics Letters, 112 (2021), 106705.  doi: 10.1016/j.aml.2020.106705.

[7]

B. PeiY. Xu and J. L. Wu, Stochastic averaging for stochastic differential equations driven by fractional Brownian motion and standard Brownian motion, Applied Mathematics Letters, 100 (2020), 106006.  doi: 10.1016/j.aml.2019.106006.

[8]

J. Cui and N. Bi, Averaging principle for neutral stochastic functional differential equations with impulses and non-Lipschitz coefficients, Statistics and Probability Letters, 163 (2020), 108775.  doi: 10.1016/j.spl.2020.108775.

[9]

J. K. LiuW. Xu and Q. Guo, Averaging principle for impulsive stochastic partial differential equations, Stoch. Dynam., 21 (2021), 2150014.  doi: 10.1142/S0219493721500143.

[10]

W. Mao and X. Mao, An averaging principle for neutral stochastic functional differential equations driven by Poisson random measure, Adv. Difference Equ., 2016 (2016), 77 pp. doi: 10.1088/1751-8113/44/33/335201.

[11]

H. M. Ahmed and Q. Zhu, The averaging principle of Hilfer fractional stochastic delay differential equations with Poisson jumps, Applied Mathematics Letters, 112 (2021), 106755.  doi: 10.1016/j.aml.2020.106755.

[12]

J. Liu and W. Xu, An averaging result for impulsive fractional neutral stochastic differential equations, Applied Mathematics Letters, 114 (2021), 106892.  doi: 10.1016/j.aml.2020.106892.

[13]

W. J. XuW. Xu and S. Zhang, The averaging principle for stochastic differential equations with Caputo fractional derivative, Applied Mathematics Letters, 93 (2019), 79-84.  doi: 10.1016/j.aml.2019.02.005.

[14]

D. F. LuoQ. X. Zhu and Z. G. Luo, An averaging principle for stochastic fractional differential equations with time delays, Applied Mathematics Letters, 105 (2020), 106290.  doi: 10.1016/j.aml.2020.106290.

[15]

M. HannabouK. Hilal and A. Kajouni, Existence and uniqueness of mild solutions to impulsive nonlocal Cauchy problems, Journal of Mathematics, 2020 (2020), 5729128. 

[16]

L. Shen and J. T. Sun, Global existence of solutions for stochastic impulsive differential equations, Acta Mathematica Sinica, 27 (2011), 773-780.  doi: 10.1007/s10114-011-8650-9.

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