Deep Learning-Based Approximation of Solutions to Stochastic Differential Equations
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171-183
Digital Object Identifier:
10.58578/ijemt.v4i2.10241
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Authors:Rishav Jha1*, Kameshwar Sahani2, Suresh Kumar Sahani3, Ravi Kumar Raj4, Dilip Kumar Sah5
Copyright :Authors retain copyright and grant the journal right of first publication.
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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.
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