Markov Chains, Monte Carlo Methods, and Numerical Simulations

To accurately simulate complex systems such as weather patterns or financial markets, advanced computational techniques are essential. Markov chains provide a powerful stochastic modeling approach widely applied in statistics, physics, and computer science, where future states depend only on the current state. Monte Carlo methods complement this by using random sampling to approximate complex systems and derive statistical outcomes through repeated simulations. These techniques can be effectively implemented in MATLAB using built-in functions like rand for random number generation and matrix operations for state transitions. For Markov chain simulations, developers typically construct transition probability matrices and iterate through state evolutions using matrix multiplication. Monte Carlo implementations often involve loop structures for multiple trials and statistical analysis functions like mean or histogram for result aggregation. By leveraging MATLAB's computational capabilities, researchers can model system dynamics, analyze convergence properties, and make data-driven decisions based on robust numerical evidence.