A Fractional Optimal Control Model with Multi-Degree of Freedom for a Simple Cash Balance Problem

This study introduces a fractional optimal control model for managing cash balances, emphasizing the critical factor of long-term memory, particularly in the context of multiple investments. Leveraging the Caputo-type fractional derivative, our model captures memory effects, providing a nuanced perspective on financial dynamics. In contrast to single-degree-of-freedom fractional models, our approach assigns distinct parameters to represent memory in cash and investment behavior within a multi-degree-of-freedom framework. These parameters reflect the varying influence strengths of historical states on their corresponding current states. By doing so, we enhance the model’s flexibility and accuracy, enabling it to more faithfully depict real-world complexities. Our results reveal that introducing long-term memory yields smoother fluctuations in both cash and investment. Furthermore, the interplay of memory effects within this multi-degree-of-freedom context significantly impacts changes in enterprise cash balance. Notably, if we tentatively view memory as decision-makers’ reliance on historical data regarding cash and various investments, the magnitude of the memory parameter may be associated with the degree of historical dependence. A smaller value indicates stronger historical dependence. However, this is only a heuristic interpretation; establishing a rigorous link between memory parameters and decision-making behavior requires further empirical investigation. While our study primarily addresses straightforward cash balance problems, the clarity of our model construction and resulting insights holds broader implications. These findings can be extended to other contexts, providing meaningful conclusions for financial decision-makers.