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All Journal Asian Journal of Science, Technology, Engineering, and Art International Journal of Education, Management, and Technology African Multidisciplinary Journal of Sciences and Artificial Intelligence Open Access DRIVERset
Dilip Kumar Sah
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Agriculture, Biological Sciences & Forestry Arts Biochemistry, Genetics & Molecular Biology Chemical Engineering, Chemistry & Bioengineering Computer Science & IT Education Electrical & Electronics Engineering Engineering Environmental Science Materials Science & Nanotechnology Social Sciences

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Adaptive Time Stepping Numerical Schemes for Stochastic Differential Equations Rishav Jha; Kameshwar Sahani; Suresh Kumar Sahani; Ravi Kumar Raj; Dilip Kumar Sah
Asian Journal of Science, Technology, Engineering, and Art Vol 4 No 3 (2026): Asian Journal of Science, Technology, Engineering, and Art
Publisher : Darul Yasin Al Sys

Show Abstract | Download Original | Original Source | Check in Google Scholar | DOI: 10.58578/ajstea.v4i3.10239

Abstract

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.
Data-Driven Identification of Stochastic Dynamical Systems Rishav Jha; Kameshwar Sahani; Suresh Kumar Sahani; Ravi Kumar Raj; Dilip Kumar Sah
African Multidisciplinary Journal of Sciences and Artificial Intelligence Vol 3 No 2 (2026): African Multidisciplinary Journal of Sciences and Artificial Intelligence
Publisher : Darul Yasin Al Sys

Show Abstract | Download Original | Original Source | Check in Google Scholar | DOI: 10.58578/amjsai.v3i2.10238

Abstract

Identifying stochastic dynamical systems from observational data remains a major challenge in applied mathematics and engineering, particularly when complex systems are influenced by random perturbations and incomplete empirical information. This comprehensive review aims to examine state-of-the-art data-driven methods for discovering governing equations, estimating parameters, and predicting the behavior of stochastic dynamical systems. The review systematically analyzes key methodological approaches, including Sparse Identification of Nonlinear Dynamics (SINDy), Dynamic Mode Decomposition (DMD) and its extensions, Koopman operator theory, neural ordinary differential equations, and Bayesian inference. Each approach is evaluated in terms of its theoretical foundations, computational requirements, robustness to noise, and applicability to different classes of stochastic systems. Drawing on numerical experiments and real-world case studies, the findings show that no single method consistently outperforms others across all scenarios. Instead, hybrid approaches that integrate physics-informed constraints with machine learning demonstrate the strongest potential for advancing data-driven system identification. The review concludes that future research should address real-time identification, uncertainty quantification, and the integration of multi-fidelity data sources to improve the reliability and scalability of stochastic system modeling. This work contributes a comprehensive framework for guiding researchers and practitioners in selecting and implementing appropriate identification methods for stochastic dynamical systems.
Data-Driven Identification of Stochastic Dynamical Systems Rishav Jha; Kameshwar Sahani; Suresh Kumar Sahani; Ravi Kumar Raj; Dilip Kumar Sah
African Multidisciplinary Journal of Sciences and Artificial Intelligence Vol 3 No 2 (2026): African Multidisciplinary Journal of Sciences and Artificial Intelligence
Publisher : Darul Yasin Al Sys

Show Abstract | Download Original | Original Source | Check in Google Scholar | DOI: 10.58578/amjsai.v3i2.10238

Abstract

Identifying stochastic dynamical systems from observational data remains a major challenge in applied mathematics and engineering, particularly when complex systems are influenced by random perturbations and incomplete empirical information. This comprehensive review aims to examine state-of-the-art data-driven methods for discovering governing equations, estimating parameters, and predicting the behavior of stochastic dynamical systems. The review systematically analyzes key methodological approaches, including Sparse Identification of Nonlinear Dynamics (SINDy), Dynamic Mode Decomposition (DMD) and its extensions, Koopman operator theory, neural ordinary differential equations, and Bayesian inference. Each approach is evaluated in terms of its theoretical foundations, computational requirements, robustness to noise, and applicability to different classes of stochastic systems. Drawing on numerical experiments and real-world case studies, the findings show that no single method consistently outperforms others across all scenarios. Instead, hybrid approaches that integrate physics-informed constraints with machine learning demonstrate the strongest potential for advancing data-driven system identification. The review concludes that future research should address real-time identification, uncertainty quantification, and the integration of multi-fidelity data sources to improve the reliability and scalability of stochastic system modeling. This work contributes a comprehensive framework for guiding researchers and practitioners in selecting and implementing appropriate identification methods for stochastic dynamical systems.
Deep Learning-Based Approximation of Solutions to Stochastic Differential Equations Rishav Jha; Kameshwar Sahani; Suresh Kumar Sahani; Ravi Kumar Raj; Dilip Kumar Sah
International Journal of Education, Management, and Technology Vol 4 No 2 (2026): International Journal of Education, Management, and Technology
Publisher : Darul Yasin Al Sys

Show Abstract | Download Original | Original Source | Check in Google Scholar | DOI: 10.58578/ijemt.v4i2.10241

Abstract

Stochastic differential equations (SDEs) are essential mathematical tools for modeling systems subject to random influences across finance, physics, biology, and engineering. However, traditional numerical methods, including the Euler–Maruyama and Milstein schemes, face substantial limitations in high-dimensional settings and often require extensive Monte Carlo simulations to obtain accurate statistical estimates. This study aims to develop and evaluate a deep learning framework for approximating SDE solutions using Physics-Informed Neural Networks (PINNs) and Deep Backward Stochastic Differential Equation methods. The proposed methodology leverages automatic differentiation to enforce the underlying stochastic dynamics through a composite loss function incorporating PDE residuals, boundary conditions, and initial conditions. The framework was assessed through benchmark problems, including geometric Brownian motion, Ornstein–Uhlenbeck processes, and the Black–Scholes equation. The findings indicate that deep learning approaches achieve superior accuracy compared with traditional numerical schemes while offering substantial computational advantages, particularly for high-dimensional problems. Experimental results show that the proposed approach achieves relative errors below 1% and provides speedup factors exceeding 100 times for 50-dimensional problems compared with conventional Monte Carlo methods. The study concludes that PINNs and Deep BSDE methods offer a promising computational paradigm for solving high-dimensional stochastic differential equations efficiently and accurately. This work contributes to scientific machine learning and numerical SDE research by demonstrating the potential of deep learning-based solvers to address dimensionality-related limitations in conventional stochastic simulation methods.
Co-Authors Kameshwar Sahani Ravi Kumar Raj Rishav Jha Suresh Kumar Sahani