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

TL;DR

This paper introduces a new theorem for evaluating the convergence order of numerical schemes for multidimensional SDEs, and proposes two efficient methods for computing stochastic integrals in the Milstein method, validated through simulations.

Contribution

It presents a novel theorem for convergence evaluation and two new techniques for stochastic integral computation in the Milstein method for multidimensional SDEs.

Findings

The proposed methods improve computational efficiency.

Simulation results demonstrate enhanced convergence rates.

Applications to financial models validate the approaches.

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.