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Short description: Randomly determined process

Stochastic (/stəˈkæstɪk/; from grc στόχος (stókhos) 'aim, guess')[1] refers to the property of being well-described by a random probability distribution.[1] Although stochasticity and randomness are distinct in that the former refers to a modeling approach and the latter refers to phenomena themselves, these two terms are often used synonymously. Furthermore, in probability theory, the formal concept of a stochastic process is also referred to as a random process.[2][3][4][5][6]

Stochasticity is used in many different fields, including the natural sciences such as biology,[7] chemistry,[8] ecology,[9] neuroscience,[10] and physics,[11] as well as technology and engineering fields such as image processing, signal processing,[12] information theory,[13] computer science,[14] cryptography,[15] and telecommunications.[16] It is also used in finance, due to seemingly random changes in financial markets[17][18][19] as well as in medicine, linguistics, music, media, colour theory, botany, manufacturing, and geomorphology.


The word stochastic in English was originally used as an adjective with the definition "pertaining to conjecturing", and stemming from a Greek word meaning "to aim at a mark, guess", and the Oxford English Dictionary gives the year 1662 as its earliest occurrence.[1] In his work on probability Ars Conjectandi, originally published in Latin in 1713, Jakob Bernoulli used the phrase "Ars Conjectandi sive Stochastice", which has been translated to "the art of conjecturing or stochastics".[20] This phrase was used, with reference to Bernoulli, by Ladislaus Bortkiewicz,[21] who in 1917 wrote in German the word Stochastik with a sense meaning random. The term stochastic process first appeared in English in a 1934 paper by Joseph Doob.[1] For the term and a specific mathematical definition, Doob cited another 1934 paper, where the term stochastischer Prozeß was used in German by Aleksandr Khinchin,[22][23] though the German term had been used earlier in 1931 by Andrey Kolmogorov.[24]


In the early 1930s, Aleksandr Khinchin gave the first mathematical definition of a stochastic process as a family of random variables indexed by the real line.[25][22][lower-alpha 1] Further fundamental work on probability theory and stochastic processes was done by Khinchin as well as other mathematicians such as Andrey Kolmogorov, Joseph Doob, William Feller, Maurice Fréchet, Paul Lévy, Wolfgang Doeblin, and Harald Cramér.[27][28] Decades later Cramér referred to the 1930s as the "heroic period of mathematical probability theory".[28]

In mathematics, the theory of stochastic processes is an important contribution to probability theory,[29] and continues to be an active topic of research for both theory and applications.[30][31][32]

The word stochastic is used to describe other terms and objects in mathematics. Examples include a stochastic matrix, which describes a stochastic process known as a Markov process, and stochastic calculus, which involves differential equations and integrals based on stochastic processes such as the Wiener process, also called the Brownian motion process.

Natural science

One of the simplest continuous-time stochastic processes is Brownian motion. This was first observed by botanist Robert Brown while looking through a microscope at pollen grains in water.


The Monte Carlo method is a stochastic method popularized by physics researchers Stanisław Ulam, Enrico Fermi, John von Neumann, and Nicholas Metropolis.[33] The use of randomness and the repetitive nature of the process are analogous to the activities conducted at a casino. Methods of simulation and statistical sampling generally did the opposite: using simulation to test a previously understood deterministic problem. Though examples of an "inverted" approach do exist historically, they were not considered a general method until the popularity of the Monte Carlo method spread.

Perhaps the most famous early use was by Enrico Fermi in 1930, when he used a random method to calculate the properties of the newly discovered neutron. Monte Carlo methods were central to the simulations required for the Manhattan Project, though they were severely limited by the computational tools of the time. Therefore, it was only after electronic computers were first built (from 1945 on) that Monte Carlo methods began to be studied in depth. In the 1950s they were used at Los Alamos for early work relating to the development of the hydrogen bomb, and became popularized in the fields of physics, physical chemistry, and operations research. The RAND Corporation and the U.S. Air Force were two of the major organizations responsible for funding and disseminating information on Monte Carlo methods during this time, and they began to find a wide application in many different fields.

Uses of Monte Carlo methods require large amounts of random numbers, and it was their use that spurred the development of pseudorandom number generators, which were far quicker to use than the tables of random numbers which had been previously used for statistical sampling.


Stochastic resonance: In biological systems, introducing stochastic "noise" has been found to help improve the signal strength of the internal feedback loops for balance and other vestibular communication.[34] It has been found to help diabetic and stroke patients with balance control.[35] Many biochemical events also lend themselves to stochastic analysis. Gene expression, for example, has a stochastic component through the molecular collisions—as during binding and unbinding of RNA polymerase to a gene promoter—via the solution's Brownian motion.


Simonton (2003, Psych Bulletin) argues that creativity in science (of scientists) is a constrained stochastic behaviour such that new theories in all sciences are, at least in part, the product of a stochastic process.

Computer science

Stochastic ray tracing is the application of Monte Carlo simulation to the computer graphics ray tracing algorithm. "Distributed ray tracing samples the integrand at many randomly chosen points and averages the results to obtain a better approximation. It is essentially an application of the Monte Carlo method to 3D computer graphics, and for this reason is also called Stochastic ray tracing."[citation needed]

Stochastic forensics analyzes computer crime by viewing computers as stochastic steps.

In artificial intelligence, stochastic programs work by using probabilistic methods to solve problems, as in simulated annealing, stochastic neural networks, stochastic optimization, genetic algorithms, and genetic programming. A problem itself may be stochastic as well, as in planning under uncertainty.


The financial markets use stochastic models to represent the seemingly random behaviour of various financial assets, including the random behavior of the price of one currency compared to that of another (such as the price of US Dollar compared to that of the Euro), and also to represent random behaviour of interest rates. These models are then used by quantitative analysts to value options on stock prices, bond prices, and on interest rates, see Markov models. Moreover, it is at the heart of the insurance industry.


The formation of river meanders has been analyzed as a stochastic process.

Language and linguistics

Non-deterministic approaches in language studies are largely inspired by the work of Ferdinand de Saussure, for example, in functionalist linguistic theory, which argues that competence is based on performance.[36][37] This distinction in functional theories of grammar should be carefully distinguished from the langue and parole distinction. To the extent that linguistic knowledge is constituted by experience with language, grammar is argued to be probabilistic and variable rather than fixed and absolute. This conception of grammar as probabilistic and variable follows from the idea that one's competence changes in accordance with one's experience with language. Though this conception has been contested,[38] it has also provided the foundation for modern statistical natural language processing[39] and for theories of language learning and change.[40]


Manufacturing processes are assumed to be stochastic processes. This assumption is largely valid for either continuous or batch manufacturing processes. Testing and monitoring of the process is recorded using a process control chart which plots a given process control parameter over time. Typically a dozen or many more parameters will be tracked simultaneously. Statistical models are used to define limit lines which define when corrective actions must be taken to bring the process back to its intended operational window.

This same approach is used in the service industry where parameters are replaced by processes related to service level agreements.


The marketing and the changing movement of audience tastes and preferences, as well as the solicitation of and the scientific appeal of certain film and television debuts (i.e., their opening weekends, word-of-mouth, top-of-mind knowledge among surveyed groups, star name recognition and other elements of social media outreach and advertising), are determined in part by stochastic modeling. A recent attempt at repeat business analysis was done by Japanese scholars[citation needed] and is part of the Cinematic Contagion Systems patented by Geneva Media Holdings, and such modeling has been used in data collection from the time of the original Nielsen ratings to modern studio and television test audiences.


Stochastic effect, or "chance effect" is one classification of radiation effects that refers to the random, statistical nature of the damage. In contrast to the deterministic effect, severity is independent of dose. Only the probability of an effect increases with dose.


In music, mathematical processes based on probability can generate stochastic elements.

Stochastic processes may be used in music to compose a fixed piece or may be produced in performance. Stochastic music was pioneered by Iannis Xenakis, who coined the term stochastic music. Specific examples of mathematics, statistics, and physics applied to music composition are the use of the statistical mechanics of gases in Pithoprakta, statistical distribution of points on a plane in Diamorphoses, minimal constraints in Achorripsis, the normal distribution in ST/10 and Atrées, Markov chains in Analogiques, game theory in Duel and Stratégie, group theory in Nomos Alpha (for Siegfried Palm), set theory in Herma and Eonta,[41] and Brownian motion in N'Shima.[citation needed] Xenakis frequently used computers to produce his scores, such as the ST series including Morsima-Amorsima and Atrées, and founded CEMAMu. Earlier, John Cage and others had composed aleatoric or indeterminate music, which is created by chance processes but does not have the strict mathematical basis (Cage's Music of Changes, for example, uses a system of charts based on the I-Ching). Lejaren Hiller and Leonard Issacson used generative grammars and Markov chains in their 1957 Illiac Suite. Modern electronic music production techniques make these processes relatively simple to implement, and many hardware devices such as synthesizers and drum machines incorporate randomization features. Generative music techniques are therefore readily accessible to composers, performers, and producers.

Social sciences

Stochastic social science theory is similar to systems theory in that events are interactions of systems, although with a marked emphasis on unconscious processes. The event creates its own conditions of possibility, rendering it unpredictable if simply for the number of variables involved. Stochastic social science theory can be seen as an elaboration of a kind of 'third axis' in which to situate human behavior alongside the traditional 'nature vs. nurture' opposition. See Julia Kristeva on her usage of the 'semiotic', Luce Irigaray on reverse Heideggerian epistemology, and Pierre Bourdieu on polythetic space for examples of stochastic social science theory.Lua error: Internal error: The interpreter exited with status 1.

The term stochastic terrorism has come into frequent use[42] with regard to lone wolf terrorism. The terms "Scripted Violence" and "Stochastic Terrorism" are linked in a "cause <> effect" relationship. "Scripted violence" rhetoric can result in an act of "stochastic terrorism". The phrase "scripted violence" has been used in social science since at least 2002.[43]

Author David Neiwert, who wrote the book Alt-America, told Salon interviewer Chauncey Devega:

Scripted violence is where a person who has a national platform describes the kind of violence that they want to be carried out. He identifies the targets and leaves it up to the listeners to carry out this violence. It is a form of terrorism. It is an act and a social phenomenon where there is an agreement to inflict massive violence on a whole segment of society. Again, this violence is led by people in high-profile positions in the media and the government. They're the ones who do the scripting, and it is ordinary people who carry it out.
Think of it like Charles Manson and his followers. Manson wrote the script; he didn't commit any of those murders. He just had his followers carry them out.[44]

Subtractive color reproduction

When color reproductions are made, the image is separated into its component colors by taking multiple photographs filtered for each color. One resultant film or plate represents each of the cyan, magenta, yellow, and black data. Color printing is a binary system, where ink is either present or not present, so all color separations to be printed must be translated into dots at some stage of the work-flow. Traditional line screens which are amplitude modulated had problems with moiré but were used until stochastic screening became available. A stochastic (or frequency modulated) dot pattern creates a sharper image.

See also


  1. Doob, when citing Khinchin, uses the term 'chance variable', which used to be an alternative term for 'random variable'.[26]

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  1. 1.0 1.1 1.2 1.3 "Stochastic". Stochastic. Oxford University Press. 
  2. Robert J. Adler; Jonathan E. Taylor (29 January 2009). Random Fields and Geometry. Springer Science & Business Media. pp. 7–8. ISBN 978-0-387-48116-6. 
  3. David Stirzaker (2005). Stochastic Processes and Models. Oxford University Press. p. 45. ISBN 978-0-19-856814-8. 
  4. Loïc Chaumont; Marc Yor (19 July 2012). Exercises in Probability: A Guided Tour from Measure Theory to Random Processes, Via Conditioning. Cambridge University Press. p. 175. ISBN 978-1-107-60655-5. 
  5. Murray Rosenblatt (1962). Random Processes. Oxford University Press. p. 91. ISBN 9780758172174. 
  6. Olav Kallenberg (8 January 2002). Foundations of Modern Probability. Springer Science & Business Media. pp. 24 and 25. ISBN 978-0-387-95313-7. 
  7. Paul C. Bressloff (22 August 2014). Stochastic Processes in Cell Biology. Springer. ISBN 978-3-319-08488-6. 
  8. N.G. Van Kampen (30 August 2011). Stochastic Processes in Physics and Chemistry. Elsevier. ISBN 978-0-08-047536-3. 
  9. Russell Lande; Steinar Engen; Bernt-Erik Sæther (2003). Stochastic Population Dynamics in Ecology and Conservation. Oxford University Press. ISBN 978-0-19-852525-7. 
  10. Carlo Laing; Gabriel J Lord (2010). Stochastic Methods in Neuroscience. OUP Oxford. ISBN 978-0-19-923507-0. 
  11. Wolfgang Paul; Jörg Baschnagel (11 July 2013). Stochastic Processes: From Physics to Finance. Springer Science & Business Media. ISBN 978-3-319-00327-6. 
  12. Edward R. Dougherty (1999). Random processes for image and signal processing. SPIE Optical Engineering Press. ISBN 978-0-8194-2513-3. 
  13. Thomas M. Cover; Joy A. Thomas (28 November 2012). Elements of Information Theory. John Wiley & Sons. p. 71. ISBN 978-1-118-58577-1. 
  14. Michael Baron (15 September 2015). Probability and Statistics for Computer Scientists, Second Edition. CRC Press. p. 131. ISBN 978-1-4987-6060-7. 
  15. Jonathan Katz; Yehuda Lindell (2007-08-31). Introduction to Modern Cryptography: Principles and Protocols. CRC Press. p. 26. ISBN 978-1-58488-586-3. 
  16. François Baccelli; Bartlomiej Blaszczyszyn (2009). Stochastic Geometry and Wireless Networks. Now Publishers Inc. pp. 200–. ISBN 978-1-60198-264-3. 
  17. J. Michael Steele (2001). Stochastic Calculus and Financial Applications. Springer Science & Business Media. ISBN 978-0-387-95016-7. 
  18. Marek Musiela; Marek Rutkowski (21 January 2006). Martingale Methods in Financial Modelling. Springer Science & Business Media. ISBN 978-3-540-26653-2. 
  19. Steven E. Shreve (3 June 2004). Stochastic Calculus for Finance II: Continuous-Time Models. Springer Science & Business Media. ISBN 978-0-387-40101-0. 
  20. O. B. Sheĭnin (2006). Theory of probability and statistics as exemplified in short dictums. NG Verlag. p. 5. ISBN 978-3-938417-40-9. 
  21. Oscar Sheynin; Heinrich Strecker (2011). Alexandr A. Chuprov: Life, Work, Correspondence. V&R unipress GmbH. p. 136. ISBN 978-3-89971-812-6. 
  22. 22.0 22.1 Doob, Joseph (1934). "Stochastic Processes and Statistics". Proceedings of the National Academy of Sciences of the United States of America 20 (6): 376–379. doi:10.1073/pnas.20.6.376. PMID 16587907. Bibcode1934PNAS...20..376D. 
  23. Khintchine, A. (1934). "Korrelationstheorie der stationeren stochastischen Prozesse". Mathematische Annalen 109 (1): 604–615. doi:10.1007/BF01449156. ISSN 0025-5831. 
  24. Kolmogoroff, A. (1931). "Über die analytischen Methoden in der Wahrscheinlichkeitsrechnung". Mathematische Annalen 104 (1): 1. doi:10.1007/BF01457949. ISSN 0025-5831. 
  25. Vere-Jones, David (2006). "Khinchin, Aleksandr Yakovlevich". Encyclopedia of Statistical Sciences. p. 4. doi:10.1002/0471667196.ess6027.pub2. ISBN 0471667196. 
  26. Snell, J. Laurie (2005). "Obituary: Joseph Leonard Doob". Journal of Applied Probability 42 (1): 251. doi:10.1239/jap/1110381384. ISSN 0021-9002. 
  27. Bingham, N. (2000). "Studies in the history of probability and statistics XLVI. Measure into probability: from Lebesgue to Kolmogorov". Biometrika 87 (1): 145–156. doi:10.1093/biomet/87.1.145. ISSN 0006-3444. 
  28. 28.0 28.1 Cramer, Harald (1976). "Half a Century with Probability Theory: Some Personal Recollections". The Annals of Probability 4 (4): 509–546. doi:10.1214/aop/1176996025. ISSN 0091-1798. 
  29. Applebaum, David (2004). "Lévy processes: From probability to finance and quantum groups". Notices of the AMS 51 (11): 1336–1347. 
  30. Jochen Blath; Peter Imkeller; Sylvie Roelly (2011). Surveys in Stochastic Processes. European Mathematical Society. pp. 5–. ISBN 978-3-03719-072-2. 
  31. Michel Talagrand (12 February 2014). Upper and Lower Bounds for Stochastic Processes: Modern Methods and Classical Problems. Springer Science & Business Media. pp. 4–. ISBN 978-3-642-54075-2. 
  32. Paul C. Bressloff (22 August 2014). Stochastic Processes in Cell Biology. Springer. pp. vii–ix. ISBN 978-3-319-08488-6. 
  33. Douglas Hubbard "How to Measure Anything: Finding the Value of Intangibles in Business" p. 46, John Wiley & Sons, 2007
  34. Hänggi, P. (2002). "Stochastic Resonance in Biology How Noise Can Enhance Detection of Weak Signals and Help Improve Biological Information Processing". ChemPhysChem 3 (3): 285–90. doi:10.1002/1439-7641(20020315)3:3<285::AID-CPHC285>3.0.CO;2-A. PMID 12503175. 
  35. Priplata, A. (2006). "Noise-Enhanced Balance Control in Patients with Diabetes and Patients with Stroke". Ann Neurol 59 (1): 4–12. doi:10.1002/ana.20670. PMID 16287079. 
  36. Newmeyer, Frederick. 2001. "The Prague School and North American functionalist approaches to syntax" Journal of Linguistics 37, pp. 101–126. "Since most American functionalists adhere to this trend, I will refer to it and its practitioners with the initials 'USF'. Some of the more prominent USFs are Joan Bybee, William Croft, Talmy Givon, John Haiman, Paul Hopper, Marianne Mithun and Sandra Thompson. In its most extreme form (Hopper 1987, 1988), USF rejects the Saussurean dichotomies such as langue vs. parôle. For early interpretivist approaches to focus, see Chomsky (1971) and Jackendoff (1972). parole and synchrony vs. diachrony. All adherents of this tendency feel that the Chomskyan advocacy of a sharp distinction between competence and performance is at best unproductive and obscurantist; at worst theoretically unmotivated. "
  37. Bybee, Joan. "Usage-based phonology." p. 213 in Darnel, Mike (ed). 1999. Functionalism and Formalism in Linguistics: General papers. John Benjamins Publishing Company
  38. Chomsky (1959). Review of Skinner's Verbal Behavior, Language, 35: 26–58
  39. Manning and Schütze, (1999) Foundations of Statistical Natural Language Processing, MIT Press. Cambridge, MA
  40. Bybee (2007) Frequency of use and the organization of language. Oxford: Oxford University Press
  41. Ilias Chrissochoidis, Stavros Houliaras, and Christos Mitsakis, "Set theory in Xenakis' EONTA", in International Symposium Iannis Xenakis, ed. Anastasia Georgaki and Makis Solomos (Athens: The National and Kapodistrian University, 2005), 241–249.
  42. Anthony Scaramucci says he does not support President Trump's reelection on YouTubeLua error: Internal error: The interpreter exited with status 1. published August 12, 2019 CNN
  43. Hamamoto, Darrell Y. (2002). "Empire of Death: Militarized Society and the Rise of Serial Killing and Mass Murder". New Political Science 24 (1): 105–120. doi:10.1080/07393140220122662. 
  44. DeVega, Chauncey (1 November 2018). "Author David Neiwert on the outbreak of political violence". Salon. 

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Further reading

  • Formalized Music: Thought and Mathematics in Composition by Iannis Xenakis, ISBN 1-57647-079-2
  • Frequency and the Emergence of Linguistic Structure by Joan Bybee and Paul Hopper (eds.), ISBN 1-58811-028-1/ISBN 90-272-2948-1 (Eur.)
  • The Stochastic Empirical Loading and Dilution Model provides documentation and computer code for modeling stochastic processes in Visual Basic for Applications.

External links

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