But often these two terms are used when the random variables are indexed by the integers or an interval of the real line. The terms stochastic process and random process are used interchangeably, often with no specific mathematical space for the set that indexes the random variables. The term random function is also used to refer to a stochastic or random process, because a stochastic process can also be interpreted as a random element in a function space. These two stochastic processes are considered the most important and central in the theory of stochastic processes, and were discovered repeatedly and independently, both before and after Bachelier and Erlang, in different settings and countries. Erlang to study the number of phone calls occurring in a certain period of time. Examples of such stochastic processes include the Wiener process or Brownian motion process, used by Louis Bachelier to study price changes on the Paris Bourse, and the Poisson process, used by A. Īpplications and the study of phenomena have in turn inspired the proposal of new stochastic processes. Furthermore, seemingly random changes in financial markets have motivated the extensive use of stochastic processes in finance. Stochastic processes have applications in many disciplines such as biology, chemistry, ecology, neuroscience, physics, image processing, signal processing, control theory, information theory, computer science, cryptography, and telecommunications. Examples include the growth of a bacterial population, an electrical current fluctuating due to thermal noise, or the movement of a gas molecule. Stochastic processes are widely used as mathematical models of systems and phenomena that appear to vary in a random manner. In probability theory and related fields, a stochastic ( / s t ə ˈ k æ s t ɪ k/) or random process is a mathematical object usually defined as a sequence of random variables, where the index of the sequence has the interpretation of time. The Wiener process is widely considered the most studied and central stochastic process in probability theory. A computer-simulated realization of a Wiener or Brownian motion process on the surface of a sphere.
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