Variable-Order Fractional Derivatives in Financial Systems

Financial systems are inherently complex, exhibiting memory effects, nonlinearity, and evolving dynamics that cannot be adequately captured by traditional differential models. This study introduces a novel financial system modeled using variable-order fractional derivatives of the Caputo-Fabrizio type, allowing the system’s memory to change dynamically over time. Three distinct memory structures-constant, periodic, and non-periodic (sigmoid-shaped)-are explored to simulate various economic regimes, such as stable markets, cyclical behaviors, and structural transitions. Through detailed numerical simulations and comparative analysis, the model demonstrates remarkable flexibility in capturing real-world financial behaviors, including oscillatory trends, amplification effects, and memory-driven regime shifts. The incorporation of variable-order dynamics provides a more adaptive and realistic framework for analyzing economic systems under uncertainty. Furthermore, the study outlines a path for integrating data-driven estimation techniques to learn the memory order $q\left(t\right)$ from empirical financial data, opening new directions for forecasting, control, and policy modeling. The proposed framework offers a significant advancement in fractional modeling, bridging theoretical innovation with practical financial relevance.