Review of the Fundamental Theory and Multidisciplinary Applications of Monte Carlo Methods

Monte Carlo methods are numerical techniques based on probability theory and random sampling, and are particularly useful for problems that are difficult to solve analytically, involve many variables, or contain substantial uncertainty. This paper reviews the theoretical foundations, general implementation procedures, and major algorithmic developments of Monte Carlo methods, with a comparative focus on two representative fields: physics and materials science, and financial risk analysis. In physics and materials research, typical applications include the Ising model, microstructural evolution, and particle transport, while financial applications mainly involve derivative pricing and investment risk assessment. The literature shows that although these fields differ in model specification and output measures, they share a common computational logic: constructing a probabilistic model, generating random samples, and estimating target quantities statistically. Direct Monte Carlo simulation, importance sampling, Markov chain Monte Carlo (MCMC), and multilevel Monte Carlo are suited to different sampling difficulties and computational requirements. Existing studies also indicate that simulation accuracy is affected by sample size, sampling efficiency, and model assumptions. By tracing methodological developments and comparing their adaptation across fields, this review provides a useful reference for the selection, understanding, and application of Monte Carlo methods in related research.