Experimental Report on Preliminary Exploration of Numerical Solution for Stochastic Differential Equations and Stochastic Integral Equations Using the Second Form of Generalized Mapping

Abstract

Traditional numerical methods for solving Stochastic Differential Equations (SDEs) and Stochastic Integral Equations (SIEs) have significant limitations: traditional methods such as the Euler-Maruyama method exhibit a "black-box" nature in modeling dynamic processes, making it difficult to separate the contributions of deterministic and stochastic components; when handling models with complex characteristics like jumps and long memory, extensions require substantial modifications to core logic, resulting in poor flexibility; they inadequately capture the generative mechanisms of stochastic processes, only outputting terminal states or statistical distributions while losing critical operational pathway information. To address these issues, this experiment explores the numerical solution of four typical models based on the second (probabilistic) form of Generalized Mapping Theory (GMT): . The experiments cover Geometric Brownian Motion (GBM), the Ornstein-Uhlenbeck (OU) process in the Heston model, jump-diffusion asset return models, and long-range dependent electricity price models. By modularly decomposing the operation set (F), object set (A), result set (B), and probability set (P), the advantages of GMT in dynamic process modeling, complex characteristic extension, and mechanism interpretability are verified. Results show that the second form of GMT can accurately reproduce the statistical properties of traditional methods while enabling visualization of operational components, supporting flexible extensions, and fully preserving the generative pathways of stochastic processes, providing an innovative mathematical tool for numerical solution of complex SDEs and SIEs.