Analysis of fractional stochastic systems driven by fractional Brownian motion with general memory kernel

Muhammad Imran Liaqat; Ali Akgül; J. Alberto Conejero; Muhammad Imran Liaqat; Ali Akgül; J. Alberto Conejero

doi:10.3934/math.2026058

AIMS Mathematics

2026, Volume 11, Issue 1: 1354-1381. doi: 10.3934/math.2026058

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Analysis of fractional stochastic systems driven by fractional Brownian motion with general memory kernel

1.
Abdus Salam School of Mathematical Sciences, Government College University, 68-B, New MuslimTown, Lahore 54600, Pakistan; Email: imran_liaqat_22@sms.edu.pk
2.
Siirt University, Art and Science Faculty, Department of Mathematics, 56100 Siirt, Turkey; Email: aliakgul00727@gmail.com
3.
Instituto Universitario de Matemática Pura y Aplicada. Universitat Politècnica de València, Spain

Received: 11 November 2025 Revised: 05 January 2026 Accepted: 13 January 2026 Published: 16 January 2026
MSC : 34A08, 34A07, 60G22

Fractional stochastic differential equations (FSDEs) driven by fractional Brownian motion (fBm) have attracted growing attention due to their ability to model systems exhibiting non-Markovian dynamics and long-range dependence, which naturally arise in many real-world phenomena characterized by hereditary and persistent randomness. In this work, we establish the existence and uniqueness of mild solutions using the Picard iteration technique for the case where the Hurst parameter satisfies $ {H} \in \left(\tfrac{1}{2}, 1\right) $. Moreover, we establish the approximate controllability of the systems under suitable conditions. To generalize the theoretical framework, we employ the Caputo–Katugampola fractional derivative (CKFD), thereby extending the analysis to a broader class of fractional stochastic systems.
- fractional Brownian motion,
- mild solutions,
- Picard iteration approach,
- fixed point approach
Citation: Muhammad Imran Liaqat, Ali Akgül, J. Alberto Conejero. Analysis of fractional stochastic systems driven by fractional Brownian motion with general memory kernel[J]. AIMS Mathematics, 2026, 11(1): 1354-1381. doi: 10.3934/math.2026058

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Abstract

Fractional stochastic differential equations (FSDEs) driven by fractional Brownian motion (fBm) have attracted growing attention due to their ability to model systems exhibiting non-Markovian dynamics and long-range dependence, which naturally arise in many real-world phenomena characterized by hereditary and persistent randomness. In this work, we establish the existence and uniqueness of mild solutions using the Picard iteration technique for the case where the Hurst parameter satisfies $ {H} \in \left(\tfrac{1}{2}, 1\right) $. Moreover, we establish the approximate controllability of the systems under suitable conditions. To generalize the theoretical framework, we employ the Caputo–Katugampola fractional derivative (CKFD), thereby extending the analysis to a broader class of fractional stochastic systems.

References

[1]	M. I. Liaqat, S. Etemad, S. Rezapour, A novel analytical Aboodh residual power series method for solving linear and nonlinear time-fractional partial differential equations with variable coefficients, AIMS Math., 7 (2022), 16917–16948. https://doi.org/10.3934/math.2022929 doi: 10.3934/math.2022929
[2]	S. Ghosh, An analytical approach for the fractional-order Hepatitis B model using new operator, Int. J. Biomath., 17 (2024), 2350008. https://doi.org/10.1142/S1793524523500080 doi: 10.1142/S1793524523500080
[3]	Y. Li, Y. Li, T. Yue, J. Cui, H. Wang, Effects of external magnetic field on the reflection and transmission of thermoelastic coupled waves with consideration of fractional order thermoelasticity, Mech. Adv. Mater. Struct., 32 (2025), 2381–2392. https://doi.org/10.1080/15376494.2024.2379504 doi: 10.1080/15376494.2024.2379504
[4]	M. I. Liaqat, Z. A. Khan, J. A. Conejero, A. Akgül, Revised and generalized results of averaging principles for the fractional case, Axioms, 13 (2024), 732. https://doi.org/10.3390/axioms13110732 doi: 10.3390/axioms13110732
[5]	F. Jarad, T. Abdeljawad, D. Baleanu, On the generalized fractional derivatives and their Caputo modification, J. Nonlinear Sci. Appl., 10 (2017), 2607–2619. https://doi.org/10.22436/jnsa.010.05.27 doi: 10.22436/jnsa.010.05.27
[6]	L. Ma, Y. Chen, Analysis of Caputo-Katugampola fractional differential system, The Eur. Phys. J. Plus, 139 (2024), 171. https://doi.org/10.1140/epjp/s13360-024-04949-y doi: 10.1140/epjp/s13360-024-04949-y
[7]	X. Li, P. J. Wong, Two new approximations for generalized Caputo fractional derivative and their application in solving generalized fractional sub-diffusion equations, J. Appl. Math. Comput., 69 (2023), 4689–4716. https://doi.org/10.1007/s12190-023-01944-x doi: 10.1007/s12190-023-01944-x
[8]	O. Kahouli, A. Jmal, O. Naifar, A. M. Nagy, A. Ben Makhlouf, New result for the analysis of Katugampola fractional-order systems application to identification problems, Mathematics, 10 (2022), 1814. https://doi.org/10.3390/math10111814 doi: 10.3390/math10111814
[9]	S. Xiao, J. Li, New result on finite-time stability for Caputo-Katugampola fractional-order neural networks with time delay, Neural Process. Lett., 55 (2023), 7951–7966. https://doi.org/10.1007/s11063-023-11291-4 doi: 10.1007/s11063-023-11291-4
[10]	N. H. Sweilam, A. M. Nagy, T. M. Al-Ajami, Numerical solutions of fractional optimal control with Caputo-Katugampola derivative, Adv. Differ. Equ., 2021 (2021), 425. https://doi.org/10.1186/s13662-021-03580-w doi: 10.1186/s13662-021-03580-w
[11]	N. Nazeer, M. I. Asjad, M. K. Azam, A. Akgül, Study of results of katugampola fractional derivative and Chebyshev inequailities, Int. J. Appl. Comput. Math., 8 (2022), 225. https://doi.org/10.1007/s40819-022-01426-x doi: 10.1007/s40819-022-01426-x
[12]	N. Van Hoa, H. Vu, T. M. Duc, Fuzzy fractional differential equations under Caputo-Katugampola fractional derivative approach, Fuzzy Sets Syst., 375 (2019), 70–99. https://doi.org/10.1016/j.fss.2018.08.001 doi: 10.1016/j.fss.2018.08.001
[13]	S. Zeng, D. Baleanu, Y. Bai, G. Wu, Fractional differential equations of Caputo-Katugampola type and numerical solutions, Appl. Math. Comput., 315 (2017), 549–554. https://doi.org/10.1016/j.amc.2017.07.003 doi: 10.1016/j.amc.2017.07.003
[14]	R. Almeida, A. B. Malinowska, T. Odzijewicz, Fractional differential equations with dependence on the Caputo-Katugampola derivative, J. Comput. Nonlinear Dyn., 11 (2016), 061017. https://doi.org/10.1115/1.4034432 doi: 10.1115/1.4034432
[15]	S. S. Redhwan, S. L. Shaikh, M. S. Abdo, Theory of nonlinear Caputo-Katugampola fractional differential equations, Unpublished preprint, arXiv: 1911.08884, 2019.
[16]	M. E. Omaba, H. A. Sulaimani, On Caputo-Katugampola fractional stochastic differential equation, Mathematics, 10 (2022), 2086. https://doi.org/10.3390/math10122086 doi: 10.3390/math10122086
[17]	G. Muñoz-Gil, G. Volpe, M. A. Garcia-March, E. Aghion, A. Argun, C. B. Hong, et al., Objective comparison of methods to decode anomalous diffusion, Nat. Commun., 12 (2021), 6253. https://doi.org/10.1038/s41467-021-26320-w doi: 10.1038/s41467-021-26320-w
[18]	G. Munoz-Gil, H. Bachimanchi, J. Pineda, B. Midtvedt, G. Fernández-Fernández, B. Requena, et al., Quantitative evaluation of methods to analyze motion changes in single-particle experiments, Nat. Commun., 16 (2025), 6749. https://doi.org/10.1038/s41467-025-61949-x
[19]	O. Kahouli, S. Albadran, Z. Elleuch, Y. Bouteraa, A. B. Makhlouf, Stability results for neutral fractional stochastic differential equations, AIMS Math., 9 (2024), 3253–3263. https://doi.org/10.3934/math.2024158 doi: 10.3934/math.2024158
[20]	A. Ali, K. Hayat, A. Zahir, K. Shah, T. Abdeljawad, Qualitative analysis of fractional stochastic differential equations with variable order fractional derivative, Qual. Theory Dyn. Syst., 23 (2024), 120. https://doi.org/10.1007/s12346-024-00982-5 doi: 10.1007/s12346-024-00982-5
[21]	A. Raheem, F. M. Alamrani, J. Akhtar, A. Alatawi, E. Alshaban, A. Khatoon, et al., Study on controllability for $\Psi$-Hilfer fractional stochastic differential equations, Fract. Fract., 8 (2024), 727. https://doi.org/10.3390/fractalfract8120727 doi: 10.3390/fractalfract8120727
[22]	K. Ramkumar, K. Ravikumar, S. Varshini, Fractional neutral stochastic differential equations with Caputo fractional derivative: Fractional Brownian motion, Poisson jumps, and optimal control, Stoch. Anal. Appl., 39 (2021), 157–176. https://doi.org/10.1080/07362994.2020.1789476 doi: 10.1080/07362994.2020.1789476
[23]	A. M. Djaouti, M. I. Liaqat, Qualitative analysis for the solutions of fractional stochastic differential equations, Axioms, 13 (2024), 438. https://doi.org/10.3390/axioms13070438 doi: 10.3390/axioms13070438
[24]	E. S. Aly, M. I. Liaqat, S. Alshammari, M. El-Morshedy, Well-posedness and stability of fractional stochastic integro-differential equations with general memory effects, AIMS Math., 10 (2025), 22265–22293. https://doi.org/10.3934/math.2025992 doi: 10.3934/math.2025992
[25]	M. Lavanya, B. S. Vadivoo, Analysis of controllability in Caputo-Hadamard stochastic fractional differential equations with fractional Brownian motion, Int. J. Dyn. Control, 12 (2024), 15–23. https://doi.org/10.1007/s40435-023-01244-z doi: 10.1007/s40435-023-01244-z
[26]	S. Moualkia, Y. Xu, On the existence and uniqueness of solutions for multidimensional fractional stochastic differential equations with variable order, Mathematics, 9 (2021), 2106. https://doi.org/10.3390/math9172106 doi: 10.3390/math9172106
[27]	C. Liping, M. A. Khan, A. Atangana, S. Kumar, A new financial chaotic model in Atangana-Baleanu stochastic fractional differential equations, Alex. Eng. J., 60 (2021), 5193–5204. https://doi.org/10.1016/j.aej.2021.04.023 doi: 10.1016/j.aej.2021.04.023
[28]	M. Abouagwa, F. Cheng, J. Li, Impulsive stochastic fractional differential equations driven by fractional Brownian motion, Adv. Differ. Equ., 2020 (2020), 57. https://doi.org/10.1186/s13662-020-2533-2 doi: 10.1186/s13662-020-2533-2
[29]	J. A. Asadzade, N. I. Mahmudov, Finite time stability analysis for fractional stochastic neutral delay differential equations, J. Appl. Math. Comput., 70 (2024), 5293–5317. https://doi.org/10.1007/s12190-024-02174-5 doi: 10.1007/s12190-024-02174-5
[30]	Z. He, S. Wang, J. Shi, D. Liu, X. Duan, Y. Shang, Physics-informed neural network supported wiener process for degradation modeling and reliability prediction, Reliab. Eng. Syst. Safe., 258 (2025), 110906. https://doi.org/10.1016/j.ress.2025.110906 doi: 10.1016/j.ress.2025.110906
[31]	Z. He, S. Wang, D. Liu, A nonparametric degradation modeling method based on generalized stochastic process with B- spline function and Kolmogorov hypothesis test considering distribution uncertainty, Comput. Ind. Eng., 203 (2025), 111036. https://doi.org/10.1016/j.cie.2025.111036 doi: 10.1016/j.cie.2025.111036
[32]	Z. He, S. Wang, D. Liu, A degradation modeling method based on artificial neural network supported Tweedie exponential dispersion process, Adv. Eng. Inform., 65 (2025), 103376. https://doi.org/10.1016/j.aei.2025.103376 doi: 10.1016/j.aei.2025.103376
[33]	S. J. Lin, Stochastic analysis of fractional Brownian motions, Stochastics, 55 (1995), 121–140. https://doi.org/10.1080/17442509508834021 doi: 10.1080/17442509508834021
[34]	T. Caraballo, M. J. Garrido-Atienza, T. Taniguchi, The existence and exponential behavior of solutions to stochastic delay evolution equations with a fractional Brownian motion, Nonlinear Anal., 74 (2011), 3671–3684. https://doi.org/10.1016/j.na.2011.02.047 doi: 10.1016/j.na.2011.02.047
[35]	G. Arthi, J. H. Park, H. Y. Jung, Existence and exponential stability for neutral stochastic integrodifferential equations with impulses driven by a fractional Brownian motion, Commun. Nonlinear Sci. Numer. Simul., 32 (2016), 145–157. https://doi.org/10.1016/j.cnsns.2015.08.014 doi: 10.1016/j.cnsns.2015.08.014
[36]	Y. Zhou, F. Jiao, Existence of mild solutions for fractional neutral evolution equations, Comput. Math. Appl., 59 (2010), 1063–1077. https://doi.org/10.1016/j.camwa.2009.06.026 doi: 10.1016/j.camwa.2009.06.026
[37]	K. J. Engel, R. Nagel, One-parameter semigroups for linear evolution equations, New York, NY: Springer New York, 2000.
[38]	Y. Zhou, Basic theory of fractional differential equations, World scientific, 2023. https://doi.org/10.1142/13289
[39]	K. Naito, Controllability of semilinear control systems dominated by the linear part, SIAM J. Control Optim., 25 (1987), 715–722. https://doi.org/10.1137/0325040 doi: 10.1137/0325040
[40]	N. Sukavanam, M. Kumar, S-controllability of an abstract first order semilinear control system, Numer. Funct. Anal. Optim., 31 (2010), 1023–1034. https://doi.org/10.1080/01630563.2010.498598 doi: 10.1080/01630563.2010.498598
[41]	Y. Zhou, L. Zhang, X. H. Shen, Existence of mild solutions for fractional evolution equations, Comput. Math. Appl., 25 (2013), 557–586. https://doi.org/10.1216/JIE-2013-25-4-557 doi: 10.1216/JIE-2013-25-4-557

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