What are Stochastic Processes?

A stochastic, or random process is a family of random variables in statistics and Probability Theory usually dictated by number sets, defined as time intervals indicating its interpretation and reflecting continuously changing numerical values within a random system; (e.g., water current or “tide” fluctuation, and bacterial culture growth). Stochastic processes confer a multitude of applications in modeling and rationalizing the prevalent randomity frequently encountered  among systems in fields, such as computer science, physics, neuroscience, and even ecology. Further study and research of this randomity phenomenon latent in financial markets and information theory among other fields, such as that by mathematicians Norbert Wiener, Louis Bachelier, and Agner Krarup Erlang  precipitated and inspired the conception of new Stochastic Processes, (e.g., Wiener Process or Brownian Motion process, and Poisson process). The synonymous terms Stochastic process and Random process sometimes refer to a random element in a function space in which case they can be alternatively characterized as random function. Stochastic processes consist of many distinct categories defined by their mathematical properties and application, (e.g., martingales, random walks, Markov processes, Lévy processes, renewal processes, random fields, and branching processes). The application of stochastics involves the implementation of a multitude of mathematical procedures, such as probability, set theory, linear algebra, calculus and topology in addition to methods of mathematical analysis, such as measure theory, real analysis, stochastic calculus, Fourier analysis, and functional analysis. Stochastic modeling and forecasting in finance incorporate Markov Model composed of individual Markov chains in which a transition or change to another state occurs according to the probability determined within the current state.