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Stochastic process

Part of a series on statistics
Probability theory
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 (/stəˈkæstɪk/) or random process is a mathematical object usually defined as a family of random variables in a probability space, where the index of the family 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]

  1. ^ a b c Joseph L. Doob (1990). Stochastic processes. Wiley. pp. 46, 47.
  2. ^ Cite error: The named reference RogersWilliams2000page1 was invoked but never defined (see the help page).
  3. ^ Cite error: The named reference Steele2012page29 was invoked but never defined (see the help page).
  4. ^ a b Emanuel Parzen (2015). Stochastic Processes. Courier Dover Publications. pp. 7, 8. ISBN 978-0-486-79688-8.
  5. ^ 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.
  6. ^ Bressloff, Paul C. (2014). Stochastic Processes in Cell Biology. Springer. ISBN 978-3-319-08488-6.
  7. ^ Van Kampen, N. G. (2011). Stochastic Processes in Physics and Chemistry. Elsevier. ISBN 978-0-08-047536-3.
  8. ^ Lande, Russell; Engen, Steinar; Sæther, Bernt-Erik (2003). Stochastic Population Dynamics in Ecology and Conservation. Oxford University Press. ISBN 978-0-19-852525-7.
  9. ^ Laing, Carlo; Lord, Gabriel J. (2010). Stochastic Methods in Neuroscience. Oxford University Press. ISBN 978-0-19-923507-0.
  10. ^ Paul, Wolfgang; Baschnagel, Jörg (2013). Stochastic Processes: From Physics to Finance. Springer Science+Business Media. ISBN 978-3-319-00327-6.
  11. ^ Dougherty, Edward R. (1999). Random processes for image and signal processing. SPIE Optical Engineering Press. ISBN 978-0-8194-2513-3.
  12. ^ Bertsekas, Dimitri P. (1996). Stochastic Optimal Control: The Discrete-Time Case. Athena Scientific. ISBN 1-886529-03-5.
  13. ^ Thomas M. Cover; Joy A. Thomas (2012). Elements of Information Theory. John Wiley & Sons. p. 71. ISBN 978-1-118-58577-1.
  14. ^ Baron, Michael (2015). Probability and Statistics for Computer Scientists (2nd ed.). CRC Press. p. 131. ISBN 978-1-4987-6060-7.
  15. ^ Baccelli, François; Blaszczyszyn, Bartlomiej (2009). Stochastic Geometry and Wireless Networks. Now Publishers Inc. ISBN 978-1-60198-264-3.
  16. ^ Steele, J. Michael (2001). Stochastic Calculus and Financial Applications. Springer Science+Business Media. ISBN 978-0-387-95016-7.
  17. ^ Musiela, Marek; Rutkowski, Marek (2006). Martingale Methods in Financial Modelling. Springer Science+Business Media. ISBN 978-3-540-26653-2.
  18. ^ Shreve, Steven E. (2004). Stochastic Calculus for Finance II: Continuous-Time Models. Springer Science+Business Media. ISBN 978-0-387-40101-0.
  19. ^ Iosif Ilyich Gikhman; Anatoly Vladimirovich Skorokhod (1969). Introduction to the Theory of Random Processes. Courier Corporation. ISBN 978-0-486-69387-3.
  20. ^ Murray Rosenblatt (1962). Random Processes. Oxford University Press.
  21. ^ 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.
  22. ^ 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.
  23. ^ 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.
  24. ^ 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.
  25. ^ 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.
  26. ^ Valeriy Skorokhod (2005). Basic Principles and Applications of Probability Theory. Springer Science & Business Media. p. 42. ISBN 978-3-540-26312-8.
  27. ^ a b Olav Kallenberg (2002). Foundations of Modern Probability. Springer Science & Business Media. pp. 24–25. ISBN 978-0-387-95313-7.
  28. ^ a b John Lamperti (1977). Stochastic processes: a survey of the mathematical theory. Springer-Verlag. pp. 1–2. ISBN 978-3-540-90275-1.
  29. ^ 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.
  30. ^ Robert J. Adler; Jonathan E. Taylor (2009). Random Fields and Geometry. Springer Science & Business Media. pp. 7–8. ISBN 978-0-387-48116-6.
  31. ^ Gregory F. Lawler; Vlada Limic (2010). Random Walk: A Modern Introduction. Cambridge University Press. ISBN 978-1-139-48876-1.
  32. ^ David Williams (1991). Probability with Martingales. Cambridge University Press. ISBN 978-0-521-40605-5.
  33. ^ L. C. G. Rogers; David Williams (2000). Diffusions, Markov Processes, and Martingales: Volume 1, Foundations. Cambridge University Press. ISBN 978-1-107-71749-7.
  34. ^ David Applebaum (2004). Lévy Processes and Stochastic Calculus. Cambridge University Press. ISBN 978-0-521-83263-2.
  35. ^ Mikhail Lifshits (2012). Lectures on Gaussian Processes. Springer Science & Business Media. ISBN 978-3-642-24939-6.
  36. ^ Robert J. Adler (2010). The Geometry of Random Fields. SIAM. ISBN 978-0-89871-693-1.
  37. ^ Samuel Karlin; Howard E. Taylor (2012). A First Course in Stochastic Processes. Academic Press. ISBN 978-0-08-057041-9.
  38. ^ Bruce Hajek (2015). Random Processes for Engineers. Cambridge University Press. ISBN 978-1-316-24124-0.
  39. ^ G. Latouche; V. Ramaswami (1999). Introduction to Matrix Analytic Methods in Stochastic Modeling. SIAM. ISBN 978-0-89871-425-8.
  40. ^ 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.
  41. ^ Patrick Billingsley (2008). Probability and Measure. Wiley India Pvt. Limited. ISBN 978-81-265-1771-8.
  42. ^ Pierre Brémaud (2014). Fourier Analysis and Stochastic Processes. Springer. ISBN 978-3-319-09590-5.
  43. ^ Adam Bobrowski (2005). Functional Analysis for Probability and Stochastic Processes: An Introduction. Cambridge University Press. ISBN 978-0-521-83166-6.
  44. ^ Applebaum, David (2004). "Lévy processes: From probability to finance and quantum groups". Notices of the AMS. 51 (11): 1336–1347.
  45. ^ Jochen Blath; Peter Imkeller; Sylvie Roelly (2011). Surveys in Stochastic Processes. European Mathematical Society. ISBN 978-3-03719-072-2.
  46. ^ 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.
  47. ^ Paul C. Bressloff (2014). Stochastic Processes in Cell Biology. Springer. pp. vii–ix. ISBN 978-3-319-08488-6.


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