Stochastic Processes

The theory of stochastic processes is a method of describing time-dependent physical phenomena, based on probability theory, which allows the calculation of averaged macroscopic properties. The application allows for determining basic characteristics of stochastic processes, such as the mean or covariance function, as well as observing the evolution of the probability distribution. It also enables the stationarity of the studied process to be assessed, and for processes considered stationary, spectral analysis and the determination of correlation time. The application allows for the analysis of data from outside the program and the definition of one's own stochastic processes based on elementary processes. Both processes with a continuous set of values ​​and those assuming only discrete values ​​can be generated within the program. Specifically, when constructing a custom stochastic process, the user has access to basic processes such as the Brownian process, the Poisson process, and the α-stable Lévy process. Based on the stochastic process formula provided by the user, the program will generate a set of its realizations. The resulting set of time series (i.e., trajectories) of the given stochastic process is subjected to the analysis described above. To construct the trajectory, it is necessary to generate random variables derived from the appropriate probability distribution functions. The user can familiarize themselves with the basic types of random variables (i.e., uniform, Gaussian, exponential, Poisson, and α-stable distributions). In addition to numerical calculations, the user can also explore a theoretical introduction, which briefly discusses the basic division of stochastic processes into Markov and non-Markov processes. The theoretical discussion introduces the fundamental equations governing the dynamics of stochastic systems, such as the Fokker-Planck equation and the Langevin equation. The application also includes basic mathematical concepts related to the definitions of stochastic integrals, such as the Lebesgue-Stieltjes integral and the Ito integral. The user can also become familiar with the physical understanding of such Markov processes as the Brownian particle motion, the Ornstein-Uhlenbeck process, and the one-step process (represented by the Poisson process). The program also utilizes mathematical analysis tools such as Fourier analysis, both for the probability distribution function (determining the characteristic function) and the covariance function of the stationary process (spectral density). Spectral density allows for the determination of the noise type (white or colored noise).