Advancing Hybrid Numerical Methods for Nonlinear Stochastic Differential Equations: Applications in Complex Systems

Abstract

The focus of this work is to consider composite numerical techniques for the approximation of SDEs with nonlinear coefficients in the drift and diffusion terms. SDEs, crucial for modeling systems with stochastic components, contain nonlinear terms that cause analytical solvability, numerical stiffness, and sensitivity to noise. These difficulties pose a problem for traditional techniques such as Euler-Maruyama or Milstein schemes, specifically in stiff or very nonlinear systems. Accompanying exact methods are numerical methods that include a deterministic synthesis of drift terms and a stochastic interpolation of diffusion terms with the purpose of increasing precision and stability and optimizing used computing time. Discussed approaches include implicit-explicit (IMEX) schemes, spectral collocation methods, and machine learning-assisted techniques. IMEX methods handle stiffness in nonlinear drift terms implicitly, while explicitly handling stochastic diffusion. Spectral-collocation methods utilize high-order polynomial approximations for accuracy in discretization where solutions are smooth and defined in a bounded domain. The combination of these techniques and machine learning extends SDE analysis and concentrates on SDE nonlinearities as well as adaptive solution strategies. They find use in every area of discipline, such as stochastic volatility models in finance, population dynamics in biology, and turbulent fluid flows in engineering. Simulation results show that hybrid schemes outperform other methods in terms of accuracy, stability, and computational expense. This work outlines how the integration of the suggested methods can overcome the shortcomings of the classic approaches so as to enable progression in solving complex, high-dimensional, and nonlinear stochastic problems. Subsequent studies will continue to investigate additional adaptive frameworks and more domain-specific and machine learning-based improvements to expand the spectrum of hybrid use.