Qualitative Analysis of Uncertain Fractional Differential Equations and Application to Interest Rate Modeling

Uncertain fractional differential equations model complex systems that exhibit memory effects and are influenced by human-based uncertainty. These equations provide a flexible and accurate framework for representing real-world phenomena, particularly in situations where traditional probabilistic methods are inadequate, such as modeling financial market systems where uncertainty and memory effects play a significant role. This research first presents an existence and uniqueness result for the uncertain fractional system with the φ-Hilfer fractional derivative, obtained via the successive approximation approach. Then the analytical solution is derived using the Mittag–Leffler function, and sample continuity is demonstrated under Lipschitz and linear growth conditions. To illustrate the applicability of the theory, we consider an interest rate model and provide two numerical examples to support the theoretical results on existence and uniqueness. All results are developed using the φ-Hilfer fractional derivative, which generalizes the Caputo, Hadamard, and Katugampola fractional derivatives. Consequently, the results are presented in a generalized form.