MATLAB Code Implementation for Stochastic Process Simulation

Stochastic process simulation serves as a vital tool for studying uncertain phenomena, and MATLAB offers a comprehensive function library for simulating various types of stochastic processes. This article demonstrates how to implement common stochastic process simulations using MATLAB, including generation methods for Poisson processes, exponential distributions, and Gaussian distributions. A Poisson process is a counting process frequently used to model the occurrence rate of random events. In MATLAB implementation, we can construct a Poisson process by generating inter-arrival times following an exponential distribution. The simulation approach involves first determining the average arrival rate of events, then applying the inverse transform method to generate corresponding arrival times. By accumulating these time intervals, we obtain the event sequence of the Poisson process. The key MATLAB functions for this implementation include exprnd for generating exponential random variables and cumsum for time accumulation. Exponential distribution is commonly employed to describe inter-arrival times in Poisson processes. MATLAB's built-in random number generation functions can directly produce random numbers following exponential distribution by specifying distribution parameters such as the mean or rate parameter. Through parameter adjustment, we can simulate event flows with different intensities. The exprnd function accepts the rate parameter lambda and returns exponentially distributed random numbers, enabling flexible simulation of various arrival patterns. Gaussian distribution (normal distribution) finds extensive applications in stochastic process simulation, particularly when modeling noise or random walks. MATLAB provides multiple methods for generating Gaussian random numbers, including functions for standard normal distribution and custom functions that allow specification of mean and variance parameters. By combining multiple independent Gaussian random variables, we can construct more complex stochastic processes such as Brownian motion. The randn function generates standard normal random variables, while normrnd from the Statistics and Machine Learning Toolbox enables customized normal distribution generation with specified mean and standard deviation parameters. For more advanced stochastic process simulations, we can integrate these fundamental distributions with MATLAB's matrix operation capabilities to build multidimensional or non-stationary stochastic process models. Additionally, the Statistics and Machine Learning Toolbox offers advanced functions for specialized analysis and visualization, including time series modeling tools and spectral estimation methods that enhance simulation accuracy and analytical depth.