Stochastic process

In probability theory and related fields, a stochastic (/stˈkæstɪk/) or random process is a mathematical object usually defined as a family of random variables. Stochastic processes are widely used as mathematical models of systems and phenomena that appear to vary in a random manner. Examples include the growth of a bacterial population, an electrical current fluctuating due to thermal noise, or the movement of a gas molecule.[1][4][5] Stochastic processes have applications in many disciplines such as biology,[6] chemistry,[7] ecology,[8] neuroscience,[9] physics,[10] image processing, signal processing,[11] control theory,[12] information theory,[13] computer science,[14] cryptography[15] and telecommunications.[16] Furthermore, seemingly random changes in financial markets have motivated the extensive use of stochastic processes in finance.[17][18][19]

A computer-simulated realization of a Wiener or Brownian motion process on the surface of a sphere. The Wiener process is widely considered the most studied and central stochastic process in probability theory.[1][2][3]

Applications and the study of phenomena have in turn inspired the proposal of new stochastic processes. Examples of such stochastic processes include the Wiener process or Brownian motion process,[lower-alpha 1] used by Louis Bachelier to study price changes on the Paris Bourse,[22] and the Poisson process, used by A. K. Erlang to study the number of phone calls occurring in a certain period of time.[23] These two stochastic processes are considered the most important and central in the theory of stochastic processes,[1][4][24] and were discovered repeatedly and independently, both before and after Bachelier and Erlang, in different settings and countries.[22][25]

The term random function is also used to refer to a stochastic or random process,[26][27] because a stochastic process can also be interpreted as a random element in a function space.[28][29] The terms stochastic process and random process are used interchangeably, often with no specific mathematical space for the set that indexes the random variables.[28][30] But often these two terms are used when the random variables are indexed by the integers or an interval of the real line.[5][30] If the random variables are indexed by the Cartesian plane or some higher-dimensional Euclidean space, then the collection of random variables is usually called a random field instead.[5][31] The values of a stochastic process are not always numbers and can be vectors or other mathematical objects.[5][29]

Based on their mathematical properties, stochastic processes can be grouped into various categories, which include random walks,[32] martingales,[33] Markov processes,[34] Lévy processes,[35] Gaussian processes,[36] random fields,[37] renewal processes, and branching processes.[38] The study of stochastic processes uses mathematical knowledge and techniques from probability, calculus, linear algebra, set theory, and topology[39][40][41] as well as branches of mathematical analysis such as real analysis, measure theory, Fourier analysis, and functional analysis.[42][43][44] The theory of stochastic processes is considered to be an important contribution to mathematics[45] and it continues to be an active topic of research for both theoretical reasons and applications.[46][47][48]

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