A Classic Example for Solving Systems of Stochastic Differential Equations

A classic example for solving systems of stochastic differential equations is Brownian motion, which has widespread applications in physics, mathematics, and finance. Brownian motion refers to a stochastic process characterized by random step sizes and directions in continuous time. In physics, Brownian motion can model the diffusion behavior of small particles in liquids or gases. From a mathematical perspective, Brownian motion is extensively studied for its analytical properties and limit theorems. In financial modeling, Brownian motion serves as a fundamental component for describing price movements of stocks and other financial instruments. Understanding and mastering solution methods for Brownian motion, such as Euler-Maruyama and Milstein schemes for numerical simulation, holds crucial reference value for research in these domains. Implementation typically involves discretizing time intervals and generating Wiener process increments using Gaussian random variables, with key functions including random number generation and cumulative sum calculations for path simulation.