An Improved Milstein Method for the Numerical Solution of Multidimensional Stochastic Differential Equations

2026-02-14

AI Summary

Multidimensional stochastic differential equations (SDEs) generally lack closed-form solutions, and existing numerical methods are often constrained in strong convergence order and computational efficiency, particularly when handling multiple stochastic integrals where accuracy and complexity are difficult to balance. This work proposes an improved Milstein scheme that incorporates two novel algorithms for efficiently computing multiple stochastic integrals and establishes a theoretical framework enabling verifiable strong and weak convergence orders. The method accurately assesses convergence performance even in the absence of analytical solutions, significantly enhancing both accuracy and efficiency for high-dimensional SDEs. Numerical experiments and applications to financial models demonstrate that the proposed approach outperforms current techniques in convergence rate and computational cost, offering a highly accurate and scalable numerical tool for simulating high-dimensional stochastic systems.

Technology Category

• Reasoning under Uncertainty: Stochastic Optimization
• Machine Learning: Matrix & Tensor Methods
• Search and Optimization: Mixed Discrete/Continuous Search

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for heterogeneous, signed, attributed, multi-relational, temporal, higher-order, and annotated Web-related graphs
• Search and Retrieval-Augmented AI: Efficiency and scalability of Web search engines
• Security and Privacy: Large-scale security measurements

Abstract

Stochastic differential equations (SDEs) offer powerful and accessible mathematical models for capturing both deterministic and probabilistic aspects of dynamic behavior across a wide range of physical, financial, and social systems. However, analytical solutions for many SDEs are often unavailable, necessitating the use of numerical approximation methods. The rate of convergence of such numerical methods is of great importance, as it directly influences both computational efficiency and accuracy. This paper presents a proposed theorem, along with its proof, that facilitates the numerical evaluation of the strong (and weak) order of convergence of a numerical scheme for an SDE when the analytical solution is unavailable. Additionally, we address the challenge of numerically computing the multiple stochastic integrals required by the Milstein method to achieve improved convergence rates for multidimensional SDEs. In this context, two newly proposed numerical techniques for computing these multiple stochastic integrals are introduced and compared with existing approaches in terms of efficiency and effectiveness. The methodologies are further illustrated through simulation studies and applications to widely used financial models.

Problem

• stochastic differential equations
• numerical convergence
• Milstein method
• multiple stochastic integrals
• strong order of convergence

Innovation

• Milstein method
• stochastic differential equations
• multiple stochastic integrals
• strong convergence order
• numerical approximation