Legendre spectral-Monte Carlo method and its error analysis for nonlinear stochastic Itô–Volterra integral equation

Nonlinear stochastic Itô–Volterra integral equations (NSIVIEs) represent systems whose current state is influenced by random fluctuations and is dependent on previous information. These equations appear in many real-world scenarios, including engineering systems, biological processes, financial markets, heterogeneous media, complex transport phenomena, and viscoelastic materials. Strong numerical frameworks are required because analytical solutions for these equations are rarely available, particularly when nonlinearities and random fluctuations are present. To effectively solve NSIVIEs, in this study we propose a new hybrid numerical framework that combines Monte Carlo simulation and Legendre spectral collocation. By using orthogonal polynomial basis functions to approximate the solution, this method provides spectral accuracy while handling the hereditary memory component of the Volterra equation through a high-order Legendre spectral collocation method. A precise statistical treatment of the random fluctuations is made possible by simultaneously addressing the stochastic Itô noise through Monte Carlo sampling across numerous independent realizations. We perform a thorough convergence analysis and obtain explicit error bounds that measure the decrease in approximation error with increasing spectral resolution and Monte Carlo sample count. Numerical experiments show that the method can accurately reproduce complex stochastic behaviors and validate theoretical predictions.