A Fractional Kernel Stochastic Volterra Integro-Differential Model for Hedge Fund Return Dynamics Under Memory Effects

Abstract

This study presents a Fractional Kernel Stochastic Volterra Integro-Differential Model (FK- SVIDM) for modeling hedge fund return dynamics under memory effects and long-range dependence. Integro-differential equations are a powerful tool for modeling systems where the current state depends not only on instantaneous factors but also on the history of past states (Ejes, 2026). Traditional Markovian stochastic differential equation models are limited in their ability to capture persistent shock transmission, delayed response, and long-term dependence that characterize hedge fund strategies and performance (Lo, 2001; Kosowski, Naik, & Teo, 2007). To address these limitations, the proposed framework incorporates a fractional Volterra kernel that allows past returns to influence current dynamics through a power-law decay structure, explicitly modeling memory effects (Gatheral, Jaisson, & Rosenbaum, 2018; Bayer, Friz, & Gatheral, 2016). Given the scarcity, smoothing, and reporting constraints of high-quality hedge fund return data (Getmansky, Lo, & Makarov, 2004), this study adopts a controlled synthetic data approach. Hedge fund return series are generated under varying degrees of fractional memory, enabling systematic investigation of how the memory parameter influences return persistence, volatility clustering, and autocorrelation structures. The stochastic Volterra framework integrates the fractional kernel into the return dynamics, capturing non-Markovian behavior and nonlinear feedback mechanisms absent in classical models (El Euch & Rosenbaum, 2019; Abi Jaber, Larsson, & Pulido, 2019). Numerical simulations based on the Euler–Volterra–Maruyama scheme demonstrate that the proposed model reproduces key stylized facts of hedge fund returns, including long-range dependence, persistent volatility, and slow decay of autocorrelations (Cont, 2001; Mandelbrot & Van Ness, 1968). Sensitivity analysis reveals that the memory parameter plays a critical role in shaping return dynamics, with stronger memory leading to increased persistence and delayed mean reversion. Comparative analysis against classical stochastic differential equation models highlights the superior ability of the FK-SVIDM to represent hedge fund risk exposures and dynamic behavior. The proposed framework provides a robust analytical and computational tool for hedge fund return modeling, performance evaluation, and risk management, and establishes a flexible foundation for future empirical calibration and real-data applications.

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