Poisson Process Algorithm in Stochastic Processes

The Poisson process algorithm in stochastic processes is a crucial method for simulating Poisson processes, widely applied in fields such as communications, finance, and physics. A Poisson process is a counting process that describes the arrival of random events within fixed time or space intervals, characterized by independent events with constant occurrence probability. In MATLAB, simulation of Poisson processes typically leverages the property that inter-event times follow an exponential distribution. The implementation approach involves: Parameter Configuration: First, define the intensity parameter (λ) of the Poisson process, representing the average number of events per unit time. Generating Exponential Intervals: Since inter-event times in a Poisson process obey an exponential distribution, MATLAB's random number generation functions (e.g., `exprnd`) can be used to produce random time intervals following this distribution. The cumulative sum of these intervals yields the exact event occurrence timestamps. Time Accumulation Simulation: Through iterative or vectorized methods, accumulate the generated time intervals until exceeding the predefined time range (e.g., total simulation time T). The resulting time series represents the event arrival times of the Poisson process. This method offers high computational efficiency and strictly adheres to the mathematical definition of Poisson processes. Moreover, MATLAB's matrix operation capabilities optimize simulations for large-scale events, making it suitable for stochastic process analysis in complex scenarios. For extended applications, the Poisson algorithm can integrate with other stochastic models, such as compound Poisson processes or non-homogeneous Poisson processes, to address more sophisticated real-world requirements.