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

Abstract

Small businesses often face unpredictable profit fluctuations due to market volatility, seasonal trends, and operational constraints. Traditional forecasting models, such as ARIMA or simple regression, frequently fail to capture the combined effects of historical performance, exogenous drivers, and stochastic variability. This study introduces a Stochastic Integro-Differential Profit Dynamics Model (SIDPDM) designed to provide more accurate and interpretable profit forecasts for small enterprises. The SIDPDM integrates a memory-dependent integral term, representing the influence of past profits, with exogenous variables such as marketing spend or seasonal demand, and incorporates multiplicative stochastic noise to model real-world uncertainty. The model is discretized using an Euler–Maruyama scheme, and parameters are estimated via weighted least squares to account for heteroskedasticity. Monte Carlo simulations are employed to generate probabilistic profit forecasts, allowing the derivation of prediction intervals and risk assessment metrics. Simulation studies and sample small-business data demonstrate that the SIDPDM captures underlying profit trends, memory effects, and random fluctuations effectively, providing a flexible and robust tool for operational planning, financial decision-making, and strategic forecasting. The results highlight the importance of accounting for historical memory and stochastic dynamics in small-business profit modeling.