Adaptive Time Stepping Numerical Schemes for Stochastic Differential Equations - Rishav Jha; Kameshwar Sahani; Suresh Kumar Sahani; Ravi Kumar Raj; Dilip Kumar Sah (Darul Yasin Al Sys, 2026)
This study presents a comprehensive examination of adaptive time-stepping numerical schemes for solving stochastic differential equations (SDEs), with particular attention to methods that automatically adjust step sizes based on local error estimates. The study aims to investigate the theoretical foundations, implementation strategies, convergence properties, and practical applications of adaptive numerical methods for SDEs. The Euler–Maruyama and Milstein schemes were extended through adaptive step-size control mechanisms, and their convergence behavior was analyzed through extensive numerical experiments implemented in Python. The study also provides detailed code examples, accessible explanations, and visualizations, including convergence plots, error analysis, and performance comparisons, to support practical understanding and implementation. The findings indicate that adaptive schemes substantially improve computational efficiency while maintaining required levels of accuracy. Specifically, the results show that adaptive methods can reduce computational costs by up to 60% compared with fixed-step methods for problems involving varying stiffness. The study concludes that adaptive time-stepping offers a robust and efficient strategy for numerical SDE simulation, particularly in computational settings where accuracy and efficiency must be balanced. Its contribution lies in integrating theoretical analysis, implementation guidance, and empirical performance evaluation to support researchers and practitioners in applying adaptive numerical schemes to stochastic differential equations.