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Peer-reviewedCC BY 4.0Axioms

Fractional Stochastic Difference Systems: Well-Posedness, Stability, Perturbation Analysis, and Macroeconomic Application

  • Muhammad Imran Liaqat1,2
  • Abdulaziz Khalid Alsharidi3
  1. 1Department of Mathematics, Faculty of Sciences, Sakarya University, Sakarya 54050, Turkey
  2. 2Department of Mathematics, National College of Business Administration & Economics, Lahore 54000, Pakistan
  3. 3Department of Mathematics and Statistics, College of Science, King Faisal University, Al-Ahsa 31982, Saudi Arabia
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Abstract

We investigate nonlinear Caputo–nabla fractional stochastic difference systems driven by independent Gaussian increments. The fractional summation identity yields an exact pathwise Volterra representation whose strictly causal structure establishes finite-horizon existence and pathwise uniqueness under measurability and linear-growth assumptions, without requiring a contraction condition. Under global Lipschitz assumptions, separate weighted ℓ1 and ℓ2 kernel estimates provide quantitative mean-square bounds for perturbations of initial data, drift and diffusion coefficients, and the fractional order. The dependence of these estimates on the exponential weight and the observation horizon is made explicit. We also establish residual-based Ulam-type stability estimates, finite-horizon moment bounds, and finite-step exactness of Picard iteration. For long-time analysis, we prove discrete Caputo–nabla comparison principles, establish decay of the associated scalar resolvent, and derive an exact quadratic identity for the corresponding dissipativity condition. The Mittag–Leffler stability result remains conditional on a history-dependent dissipativity inequality. Independently, a directly verifiable coefficient-level criterion establishes asymptotic mean-square stability and is applied to a three-dimensional illustrative macroeconomic model at fractional order ϑ=0.75. Numerical experiments, including a nonlinear Monte Carlo simulation with sampling uncertainty estimates, illustrate the theoretical results.

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Cite

BibTeXRIS
BibTeX
@article{liaqat2026fractional,
  title = {{Fractional Stochastic Difference Systems: Well-Posedness, Stability, Perturbation Analysis, and Macroeconomic Application}},
  author = {Muhammad Imran Liaqat and Abdulaziz Khalid Alsharidi},
  journal = {Axioms},
  year = {2026},
  month = oct,
  doi = {10.3390/axioms15110792},
  url = {https://www.mdpi.com/2075-1680/15/11/792},
  note = {CC BY 4.0},
}
RIS
TY  - JOUR
TI  - Fractional Stochastic Difference Systems: Well-Posedness, Stability, Perturbation Analysis, and Macroeconomic Application
AU  - Muhammad Imran Liaqat
AU  - Abdulaziz Khalid Alsharidi
PY  - 2026
DA  - 2026/10/22
JO  - Axioms
DO  - 10.3390/axioms15110792
UR  - https://www.mdpi.com/2075-1680/15/11/792
AB  - We investigate nonlinear Caputo–nabla fractional stochastic difference systems driven by independent Gaussian increments. The fractional summation identity yields an exact pathwise Volterra representation whose strictly causal structure establishes finite-horizon existence and pathwise uniqueness under measurability and linear-growth assumptions, without requiring a contraction condition. Under global Lipschitz assumptions, separate weighted ℓ1 and ℓ2 kernel estimates provide quantitative mean-square bounds for perturbations of initial data, drift and diffusion coefficients, and the fractional order. The dependence of these estimates on the exponential weight and the observation horizon is made explicit. We also establish residual-based Ulam-type stability estimates, finite-horizon moment bounds, and finite-step exactness of Picard iteration. For long-time analysis, we prove discrete Caputo–nabla comparison principles, establish decay of the associated scalar resolvent, and derive an exact quadratic identity for the corresponding dissipativity condition. The Mittag–Leffler stability result remains conditional on a history-dependent dissipativity inequality. Independently, a directly verifiable coefficient-level criterion establishes asymptotic mean-square stability and is applied to a three-dimensional illustrative macroeconomic model at fractional order ϑ=0.75. Numerical experiments, including a nonlinear Monte Carlo simulation with sampling uncertainty estimates, illustrate the theoretical results.
SN  - 2075-1680
ER  -