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,[a] used by Louis Bachelier to study price changes on the Paris Bourse,[21] and the Poisson process, used by A. K. Erlang to study the number of phone calls occurring in a certain period of time.[22] These two stochastic processes are considered the most important and central in the theory of stochastic processes,[1][4][23] and were invented repeatedly and independently, both before and after Bachelier and Erlang, in different settings and countries.[21][24]
The term random function is also used to refer to a stochastic or random process,[25][26] because a stochastic process can also be interpreted as a random element in a function space.[27][28] The terms stochastic process and random process are used interchangeably, often with no specific mathematical space for the set that indexes the random variables.[27][29] But often these two terms are used when the random variables are indexed by the integers or an interval of the real line.[5][29] 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][30] The values of a stochastic process are not always numbers and can be vectors or other mathematical objects.[5][28]
^Murray Rosenblatt (1962). Random Processes. Oxford University Press.
^ abJarrow, Robert; Protter, Philip (2004). "A short history of stochastic integration and mathematical finance: the early years, 1880–1970". A Festschrift for Herman Rubin. Institute of Mathematical Statistics Lecture Notes - Monograph Series. pp. 75–80. CiteSeerX10.1.1.114.632. doi:10.1214/lnms/1196285381. ISBN978-0-940600-61-4. ISSN0749-2170.
^Guttorp, Peter; Thorarinsdottir, Thordis L. (2012). "What Happened to Discrete Chaos, the Quenouille Process, and the Sharp Markov Property? Some History of Stochastic Point Processes". International Statistical Review. 80 (2): 253–268. doi:10.1111/j.1751-5823.2012.00181.x. ISSN0306-7734. S2CID80836.
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