Collection of random variables
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]
In probability theory and related fields, a stochastic () or random process is a mathematical object usually defined as a sequence of random variables in a probability space , where the index of the sequence often has the interpretation of time . 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] and telecommunications .[15] Furthermore, seemingly random changes in financial markets have motivated the extensive use of stochastic processes in finance .[16] [17] [18]
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]
Based on their mathematical properties, stochastic processes can be grouped into various categories, which include random walks ,[31] martingales ,[32] Markov processes ,[33] Lévy processes ,[34] Gaussian processes ,[35] random fields,[36] renewal processes , and branching processes .[37] The study of stochastic processes uses mathematical knowledge and techniques from probability , calculus , linear algebra , set theory , and topology [38] [39] [40] as well as branches of mathematical analysis such as real analysis , measure theory , Fourier analysis , and functional analysis .[41] [42] [43] The theory of stochastic processes is considered to be an important contribution to mathematics[44] and it continues to be an active topic of research for both theoretical reasons and applications.[45] [46] [47]
^ a b c Joseph L. Doob (1990). Stochastic processes . Wiley. pp. 46, 47.
^ Cite error: The named reference RogersWilliams2000page1
was invoked but never defined (see the help page ).
^ Cite error: The named reference Steele2012page29
was invoked but never defined (see the help page ).
^ a b Emanuel Parzen (2015). Stochastic Processes . Courier Dover Publications. pp. 7, 8. ISBN 978-0-486-79688-8 .
^ a b c d Iosif Ilyich Gikhman; Anatoly Vladimirovich Skorokhod (1969). Introduction to the Theory of Random Processes . Courier Corporation. p. 1. ISBN 978-0-486-69387-3 .
^ Bressloff, Paul C. (2014). Stochastic Processes in Cell Biology . Springer. ISBN 978-3-319-08488-6 .
^ Van Kampen, N. G. (2011). Stochastic Processes in Physics and Chemistry . Elsevier . ISBN 978-0-08-047536-3 .
^ Lande, Russell; Engen, Steinar; Sæther, Bernt-Erik (2003). Stochastic Population Dynamics in Ecology and Conservation . Oxford University Press . ISBN 978-0-19-852525-7 .
^ Laing, Carlo; Lord, Gabriel J. (2010). Stochastic Methods in Neuroscience . Oxford University Press . ISBN 978-0-19-923507-0 .
^ Paul, Wolfgang; Baschnagel, Jörg (2013). Stochastic Processes: From Physics to Finance . Springer Science+Business Media . ISBN 978-3-319-00327-6 .
^ Dougherty, Edward R. (1999). Random processes for image and signal processing . SPIE Optical Engineering Press. ISBN 978-0-8194-2513-3 .
^ Bertsekas, Dimitri P. (1996). Stochastic Optimal Control: The Discrete-Time Case . Athena Scientific. ISBN 1-886529-03-5 .
^ Thomas M. Cover; Joy A. Thomas (2012). Elements of Information Theory . John Wiley & Sons . p. 71. ISBN 978-1-118-58577-1 .
^ Baron, Michael (2015). Probability and Statistics for Computer Scientists (2nd ed.). CRC Press . p. 131. ISBN 978-1-4987-6060-7 .
^ Baccelli, François; Blaszczyszyn, Bartlomiej (2009). Stochastic Geometry and Wireless Networks . Now Publishers Inc. ISBN 978-1-60198-264-3 .
^ Steele, J. Michael (2001). Stochastic Calculus and Financial Applications . Springer Science+Business Media . ISBN 978-0-387-95016-7 .
^ Musiela, Marek; Rutkowski, Marek (2006). Martingale Methods in Financial Modelling . Springer Science+Business Media . ISBN 978-3-540-26653-2 .
^ Shreve, Steven E. (2004). Stochastic Calculus for Finance II: Continuous-Time Models . Springer Science+Business Media . ISBN 978-0-387-40101-0 .
^ Iosif Ilyich Gikhman; Anatoly Vladimirovich Skorokhod (1969). Introduction to the Theory of Random Processes . Courier Corporation. ISBN 978-0-486-69387-3 .
^ Murray Rosenblatt (1962). Random Processes . Oxford University Press.
^ a b Jarrow, 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. CiteSeerX 10.1.1.114.632 . doi :10.1214/lnms/1196285381 . ISBN 978-0-940600-61-4 . ISSN 0749-2170 .
^ Stirzaker, David (2000). "Advice to Hedgehogs, or, Constants Can Vary". The Mathematical Gazette . 84 (500): 197–210. doi :10.2307/3621649 . ISSN 0025-5572 . JSTOR 3621649 . S2CID 125163415 .
^ Donald L. Snyder; Michael I. Miller (2012). Random Point Processes in Time and Space . Springer Science & Business Media. p. 32. ISBN 978-1-4612-3166-0 .
^ 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 . ISSN 0306-7734 . S2CID 80836 .
^ Gusak, Dmytro; Kukush, Alexander; Kulik, Alexey; Mishura, Yuliya ; Pilipenko, Andrey (2010). Theory of Stochastic Processes: With Applications to Financial Mathematics and Risk Theory . Springer Science & Business Media. p. 21. ISBN 978-0-387-87862-1 .
^ Valeriy Skorokhod (2005). Basic Principles and Applications of Probability Theory . Springer Science & Business Media. p. 42. ISBN 978-3-540-26312-8 .
^ a b Olav Kallenberg (2002). Foundations of Modern Probability . Springer Science & Business Media. pp. 24–25. ISBN 978-0-387-95313-7 .
^ a b John Lamperti (1977). Stochastic processes: a survey of the mathematical theory . Springer-Verlag. pp. 1–2. ISBN 978-3-540-90275-1 .
^ a b Loïc Chaumont; Marc Yor (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 .
^ Robert J. Adler; Jonathan E. Taylor (2009). Random Fields and Geometry . Springer Science & Business Media. pp. 7–8. ISBN 978-0-387-48116-6 .
^ Gregory F. Lawler; Vlada Limic (2010). Random Walk: A Modern Introduction . Cambridge University Press. ISBN 978-1-139-48876-1 .
^ David Williams (1991). Probability with Martingales . Cambridge University Press. ISBN 978-0-521-40605-5 .
^ L. C. G. Rogers; David Williams (2000). Diffusions, Markov Processes, and Martingales: Volume 1, Foundations . Cambridge University Press. ISBN 978-1-107-71749-7 .
^ David Applebaum (2004). Lévy Processes and Stochastic Calculus . Cambridge University Press. ISBN 978-0-521-83263-2 .
^ Mikhail Lifshits (2012). Lectures on Gaussian Processes . Springer Science & Business Media. ISBN 978-3-642-24939-6 .
^ Robert J. Adler (2010). The Geometry of Random Fields . SIAM. ISBN 978-0-89871-693-1 .
^ Samuel Karlin; Howard E. Taylor (2012). A First Course in Stochastic Processes . Academic Press. ISBN 978-0-08-057041-9 .
^ Bruce Hajek (2015). Random Processes for Engineers . Cambridge University Press. ISBN 978-1-316-24124-0 .
^ G. Latouche; V. Ramaswami (1999). Introduction to Matrix Analytic Methods in Stochastic Modeling . SIAM. ISBN 978-0-89871-425-8 .
^ D.J. Daley; David Vere-Jones (2007). An Introduction to the Theory of Point Processes: Volume II: General Theory and Structure . Springer Science & Business Media. ISBN 978-0-387-21337-8 .
^ Patrick Billingsley (2008). Probability and Measure . Wiley India Pvt. Limited. ISBN 978-81-265-1771-8 .
^ Pierre Brémaud (2014). Fourier Analysis and Stochastic Processes . Springer. ISBN 978-3-319-09590-5 .
^ Adam Bobrowski (2005). Functional Analysis for Probability and Stochastic Processes: An Introduction . Cambridge University Press. ISBN 978-0-521-83166-6 .
^ Applebaum, David (2004). "Lévy processes: From probability to finance and quantum groups". Notices of the AMS . 51 (11): 1336–1347.
^ Jochen Blath; Peter Imkeller; Sylvie Roelly (2011). Surveys in Stochastic Processes . European Mathematical Society. ISBN 978-3-03719-072-2 .
^ Michel Talagrand (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 .
^ Paul C. Bressloff (2014). Stochastic Processes in Cell Biology . Springer. pp. vii–ix. ISBN 978-3-319-08488-6 .
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