Editing Stochastic process (section)

{{Short description|Collection of random variables}}
{{Probability fundamentals}}
[[File:BMonSphere.jpg|thumb|A computer-simulated realization of a [[Wiener process|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.<ref name="doob1953stochasticP46to47"/><ref name="RogersWilliams2000page1"/><ref name="Steele2012page29"/>]]

In [[probability theory]] and related fields, a '''stochastic''' ({{IPAc-en|s|t|ə|ˈ|k|æ|s|t|ɪ|k}}) or '''random process''' is a [[mathematical object]] usually defined as a [[Indexed family|family]] of [[random variable]]s in a [[probability space]], where the [[Index set|index]] of the family often has the interpretation of [[time]]. [[Stochastic]] processes are widely used as [[mathematical model]]s of systems and phenomena that appear to vary in a random manner. Examples include the growth of a [[bacteria]]l population, an [[electrical current]] fluctuating due to [[thermal noise]], or the movement of a [[gas]] [[molecule]].<ref name="doob1953stochasticP46to47">{{cite book|author=Joseph L. Doob|title=Stochastic processes|url=https://books.google.com/books?id=7Bu8jgEACAAJ|year=1990|publisher=Wiley|pages=46, 47}}</ref><ref name="Parzen1999">{{cite book|author=Emanuel Parzen|title=Stochastic Processes|url=https://books.google.com/books?id=0mB2CQAAQBAJ|year= 2015|publisher=Courier Dover Publications|isbn=978-0-486-79688-8|pages=7, 8}}</ref><ref name="GikhmanSkorokhod1969page1">{{cite book|author1=Iosif Ilyich Gikhman|author2=Anatoly Vladimirovich Skorokhod|title=Introduction to the Theory of Random Processes|url=https://books.google.com/books?id=q0lo91imeD0C|year=1969|publisher=Courier Corporation|isbn=978-0-486-69387-3|page=1}}</ref> Stochastic processes have applications in many disciplines such as [[biology]],<ref name="Bressloff2014">{{cite book |first=Paul C.|last=Bressloff |author-link=Paul Bressloff |title=Stochastic Processes in Cell Biology |url=https://books.google.com/books?id=SwZYBAAAQBAJ |year=2014 |publisher=Springer |isbn=978-3-319-08488-6}}</ref> [[chemistry]],<ref name="Kampen2011">{{cite book |first=N. G.|last=Van Kampen |title=Stochastic Processes in Physics and Chemistry |url=https://books.google.com/books?id=N6II-6HlPxEC |year=2011 |publisher=[[Elsevier]] |isbn=978-0-08-047536-3}}</ref> [[ecology]],<ref name="LandeEngen2003">{{cite book |first1=Russell|last1=Lande |first2=Steinar|last2=Engen |first3=Bernt-Erik|last3=Sæther |title=Stochastic Population Dynamics in Ecology and Conservation |url=https://books.google.com/books?id=6KClauq8OekC |year=2003 |publisher=[[Oxford University Press]] |isbn=978-0-19-852525-7}}</ref> [[neuroscience]],<ref name="LaingLord2010">{{cite book |first1=Carlo|last1=Laing |first2=Gabriel J.|last2=Lord |title=Stochastic Methods in Neuroscience |url=https://books.google.com/books?id=RaYSDAAAQBAJ |year=2010 |publisher=[[Oxford University Press]] |isbn=978-0-19-923507-0}}</ref> [[physics]],<ref name="PaulBaschnagel2013">{{cite book |first1=Wolfgang|last1=Paul |first2=Jörg|last2=Baschnagel |title=Stochastic Processes: From Physics to Finance |url=https://books.google.com/books?id=OWANAAAAQBAJ |year=2013 |publisher=[[Springer Science+Business Media]] |isbn=978-3-319-00327-6}}</ref> [[image processing]], [[signal processing]],<ref name="Dougherty1999">{{cite book |first=Edward R.|last=Dougherty |title=Random processes for image and signal processing |url=https://books.google.com/books?id=ePxDAQAAIAAJ |year=1999 |publisher=[[SPIE]] Optical Engineering Press |isbn=978-0-8194-2513-3}}</ref> [[stochastic control|control theory]],<ref name="Bertsekas1996">{{cite book |first=Dimitri P.|last=Bertsekas |author-link=Dimitri Bertsekas |title=Stochastic Optimal Control: The Discrete-Time Case |url=https://athenasc.com/socbook.html |year=1996 |publisher=Athena Scientific |isbn=1-886529-03-5}}</ref> [[information theory]],<ref name="CoverThomas2012page71">{{cite book |author1=Thomas M. Cover |author2=Joy A. Thomas |title=Elements of Information Theory |url=https://books.google.com/books?id=VWq5GG6ycxMC |year=2012 |publisher=[[Wiley (publisher)|John Wiley & Sons]] |isbn=978-1-118-58577-1 |page=71}}</ref> [[computer science]],<ref name="Baron2015">{{cite book |first=Michael|last=Baron |title=Probability and Statistics for Computer Scientists |edition=2nd |url=https://books.google.com/books?id=CwQZCwAAQBAJ |year=2015 |publisher=[[CRC Press]] |isbn=978-1-4987-6060-7 |page=131}}</ref> and [[telecommunications]].<ref name="BaccelliBlaszczyszyn2009">{{cite book |first1=François|last1=Baccelli |first2=Bartlomiej|last2=Blaszczyszyn |title=Stochastic Geometry and Wireless Networks |url=https://books.google.com/books?id=H3ZkTN2pYS4C |year=2009 |publisher=[[Now Publishers]] Inc. |isbn=978-1-60198-264-3}}</ref> Furthermore, seemingly random changes in [[financial market]]s have motivated the extensive use of stochastic processes in [[finance]].<ref name="Steele2001">{{cite book |first=J. Michael|last=Steele |title=Stochastic Calculus and Financial Applications |url=https://books.google.com/books?id=H06xzeRQgV4C |year=2001 |publisher=[[Springer Science+Business Media]] |isbn=978-0-387-95016-7}}</ref><ref name="MusielaRutkowski2006">{{cite book |first1=Marek|last1=Musiela |first2=Marek|last2=Rutkowski |title=Martingale Methods in Financial Modelling |url=https://books.google.com/books?id=iojEts9YAxIC |year= 2006 |publisher=[[Springer Science+Business Media]] |isbn=978-3-540-26653-2}}</ref><ref name="Shreve2004">{{cite book |first=Steven E.|last=Shreve |title=Stochastic Calculus for Finance II: Continuous-Time Models |url=https://books.google.com/books?id=O8kD1NwQBsQC |year=2004 |publisher=[[Springer Science+Business Media]] |isbn=978-0-387-40101-0}}</ref>

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,{{efn|The term ''Brownian motion'' can refer to the physical process, also known as ''Brownian movement'', and the stochastic process, a mathematical object, but to avoid ambiguity this article uses the terms ''Brownian motion process'' or ''Wiener process'' for the latter in a style similar to, for example, [[Iosif Gikhman|Gikhman]] and [[Anatoliy Skorokhod|Skorokhod]]<ref name="GikhmanSkorokhod1969">{{cite book|author1=Iosif Ilyich Gikhman|author2=Anatoly Vladimirovich Skorokhod|title=Introduction to the Theory of Random Processes|url=https://books.google.com/books?id=yJyLzG7N7r8C|year=1969|publisher=Courier Corporation|isbn=978-0-486-69387-3}}</ref> or Rosenblatt.<ref name="Rosenblatt1962">{{cite book|author=Murray Rosenblatt|title=Random Processes|url=https://archive.org/details/randomprocesses00rose_0|url-access=registration|year=1962|publisher=Oxford University Press}}</ref>}} used by [[Louis Bachelier]] to study price changes on the [[Paris Bourse]],<ref name="JarrowProtter2004">{{cite book|last1=Jarrow|first1=Robert|title=A Festschrift for Herman Rubin|last2=Protter|first2=Philip|chapter=A short history of stochastic integration and mathematical finance: the early years, 1880–1970|year=2004|pages=75–80|issn=0749-2170|doi=10.1214/lnms/1196285381|citeseerx=10.1.1.114.632|series=Institute of Mathematical Statistics Lecture Notes - Monograph Series|isbn=978-0-940600-61-4}}</ref> and the [[Poisson process]], used by [[A. K. Erlang]] to study the number of phone calls occurring in a certain period of time.<ref name="Stirzaker2000">{{cite journal|last1=Stirzaker|first1=David|title=Advice to Hedgehogs, or, Constants Can Vary|journal=The Mathematical Gazette|volume=84|issue=500|year=2000|pages=197–210|issn=0025-5572|doi=10.2307/3621649|jstor=3621649|s2cid=125163415}}</ref> These two stochastic processes are considered the most important and central in the theory of stochastic processes,<ref name="doob1953stochasticP46to47"/><ref name="Parzen1999"/><ref>{{cite book|author1=Donald L. Snyder|author2=Michael I. Miller|title=Random Point Processes in Time and Space|url=https://books.google.com/books?id=c_3UBwAAQBAJ|year=2012|publisher=Springer Science & Business Media|isbn=978-1-4612-3166-0|page=32}}</ref> and were invented repeatedly and independently, both before and after Bachelier and Erlang, in different settings and countries.<ref name="JarrowProtter2004"/><ref name="GuttorpThorarinsdottir2012">{{cite journal|last1=Guttorp|first1=Peter|last2=Thorarinsdottir|first2=Thordis L.|title=What Happened to Discrete Chaos, the Quenouille Process, and the Sharp Markov Property? Some History of Stochastic Point Processes|journal=International Statistical Review|volume=80|issue=2|year=2012|pages=253–268|issn=0306-7734|doi=10.1111/j.1751-5823.2012.00181.x|s2cid=80836 }}</ref>

The term '''random function''' is also used to refer to a stochastic or random process,<ref name="GusakKukush2010page21">{{cite book|first1=Dmytro|last1=Gusak|first2=Alexander|last2=Kukush|first3=Alexey|last3=Kulik|first4=Yuliya|last4=Mishura|author4-link=Yuliya Mishura|first5=Andrey|last5=Pilipenko|title=Theory of Stochastic Processes: With Applications to Financial Mathematics and Risk Theory|url=https://books.google.com/books?id=8Nzn51YTbX4C|year=2010|publisher=Springer Science & Business Media|isbn=978-0-387-87862-1|page=21}}</ref><ref name="Skorokhod2005page42">{{cite book|author=Valeriy Skorokhod|title=Basic Principles and Applications of Probability Theory|url=https://books.google.com/books?id=dQkYMjRK3fYC|year= 2005|publisher=Springer Science & Business Media|isbn=978-3-540-26312-8|page=42}}</ref> because a stochastic process can also be interpreted as a random element in a [[function space]].<ref name="Kallenberg2002page24"/><ref name="Lamperti1977page1">{{cite book|author=John Lamperti|title=Stochastic processes: a survey of the mathematical theory|url=https://books.google.com/books?id=Pd4cvgAACAAJ|year=1977|publisher=Springer-Verlag|isbn=978-3-540-90275-1|pages=1–2}}</ref> The terms ''stochastic process'' and ''random process'' are used interchangeably, often with no specific [[mathematical space]] for the set that indexes the random variables.<ref name="Kallenberg2002page24">{{cite book|author=Olav Kallenberg|title=Foundations of Modern Probability|url=https://books.google.com/books?id=L6fhXh13OyMC|year=2002|publisher=Springer Science & Business Media|isbn=978-0-387-95313-7|pages=24–25}}</ref><ref name="ChaumontYor2012">{{cite book|author1=Loïc Chaumont|author2=Marc Yor|title=Exercises in Probability: A Guided Tour from Measure Theory to Random Processes, Via Conditioning|url=https://books.google.com/books?id=1dcqV9mtQloC&pg=PR4|year= 2012|publisher=Cambridge University Press|isbn=978-1-107-60655-5|page=175}}</ref> But often these two terms are used when the random variables are indexed by the [[integers]] or an [[Interval (mathematics)|interval]] of the [[real line]].<ref name="GikhmanSkorokhod1969page1"/><ref name="ChaumontYor2012"/> 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.<ref name="GikhmanSkorokhod1969page1"/><ref name="AdlerTaylor2009page7">{{cite book|author1=Robert J. Adler|author2=Jonathan E. Taylor|title=Random Fields and Geometry|url=https://books.google.com/books?id=R5BGvQ3ejloC|year=2009|publisher=Springer Science & Business Media|isbn=978-0-387-48116-6|pages=7–8}}</ref> The values of a stochastic process are not always numbers and can be vectors or other mathematical objects.<ref name="GikhmanSkorokhod1969page1"/><ref name="Lamperti1977page1"/>

Based on their mathematical properties, stochastic processes can be grouped into various categories, which include [[random walk]]s,<ref name="LawlerLimic2010">{{cite book|author1=Gregory F. Lawler|author2=Vlada Limic|title=Random Walk: A Modern Introduction|url=https://books.google.com/books?id=UBQdwAZDeOEC|year= 2010|publisher=Cambridge University Press|isbn=978-1-139-48876-1}}</ref> [[Martingale (probability theory)|martingales]],<ref name="Williams1991">{{cite book|author=David Williams|title=Probability with Martingales|url=https://books.google.com/books?id=e9saZ0YSi-AC|year=1991|publisher=Cambridge University Press|isbn=978-0-521-40605-5}}</ref> [[Markov process]]es,<ref name="RogersWilliams2000">{{cite book|author1=L. C. G. Rogers|author2=David Williams|title=Diffusions, Markov Processes, and Martingales: Volume 1, Foundations|url=https://books.google.com/books?id=W0ydAgAAQBAJ&pg=PA1|year= 2000|publisher=Cambridge University Press|isbn=978-1-107-71749-7}}</ref> [[Lévy process]]es,<ref name="ApplebaumBook2004">{{cite book|author=David Applebaum|title=Lévy Processes and Stochastic Calculus|url=https://books.google.com/books?id=q7eDUjdJxIkC|year=2004|publisher=Cambridge University Press|isbn=978-0-521-83263-2}}</ref> [[Gaussian process]]es,<ref>{{cite book|author=Mikhail Lifshits|title=Lectures on Gaussian Processes|url=https://books.google.com/books?id=03m2UxI-UYMC|year=2012|publisher=Springer Science & Business Media|isbn=978-3-642-24939-6}}</ref> random fields,<ref name="Adler2010">{{cite book|author=Robert J. Adler|title=The Geometry of Random Fields|url=https://books.google.com/books?id=ryejJmJAj28C&pg=PA1|year= 2010|publisher=SIAM|isbn=978-0-89871-693-1}}</ref> [[renewal process]]es, and [[branching process]]es.<ref name="KarlinTaylor2012">{{cite book|author1=Samuel Karlin|author2=Howard E. Taylor|title=A First Course in Stochastic Processes|url=https://books.google.com/books?id=dSDxjX9nmmMC|year= 2012|publisher=Academic Press|isbn=978-0-08-057041-9}}</ref> The study of stochastic processes uses mathematical knowledge and techniques from [[probability]], [[calculus]], [[linear algebra]], [[set theory]], and [[topology]]<ref name="Hajek2015">{{cite book|author=Bruce Hajek|title=Random Processes for Engineers|url=https://books.google.com/books?id=Owy0BgAAQBAJ|year=2015|publisher=Cambridge University Press|isbn=978-1-316-24124-0}}</ref><ref name="LatoucheRamaswami1999">{{cite book|author1=G. Latouche|author2=V. Ramaswami|title=Introduction to Matrix Analytic Methods in Stochastic Modeling|url=https://books.google.com/books?id=Kan2ki8jqzgC|year=1999|publisher=SIAM|isbn=978-0-89871-425-8}}</ref><ref name="DaleyVere-Jones2007">{{cite book|author1=D.J. Daley|author2=David Vere-Jones|title=An Introduction to the Theory of Point Processes: Volume II: General Theory and Structure|url=https://books.google.com/books?id=nPENXKw5kwcC|year= 2007|publisher=Springer Science & Business Media|isbn=978-0-387-21337-8}}</ref> as well as branches of [[mathematical analysis]] such as [[real analysis]], [[measure theory]], [[Fourier analysis]], and [[functional analysis]].<ref name="Billingsley2008">{{cite book|author=Patrick Billingsley|title=Probability and Measure|url=https://books.google.com/books?id=QyXqOXyxEeIC|year=2008|publisher=Wiley India Pvt. Limited|isbn=978-81-265-1771-8}}</ref><ref name="Brémaud2014">{{cite book|author=Pierre Brémaud|title=Fourier Analysis and Stochastic Processes|url=https://books.google.com/books?id=dP2JBAAAQBAJ&pg=PA1|year= 2014|publisher=Springer|isbn=978-3-319-09590-5}}</ref><ref name="Bobrowski2005">{{cite book|author=Adam Bobrowski|title=Functional Analysis for Probability and Stochastic Processes: An Introduction|url=https://books.google.com/books?id=q7dR3d5nqaUC|year= 2005|publisher=Cambridge University Press|isbn=978-0-521-83166-6}}</ref> The theory of stochastic processes is considered to be an important contribution to mathematics<ref name="Applebaum2004">{{cite journal|last1=Applebaum|first1=David|title=Lévy processes: From probability to finance and quantum groups|journal=Notices of the AMS|volume=51|issue=11|year=2004|pages=1336–1347}}</ref> and it continues to be an active topic of research for both theoretical reasons and applications.<ref name="BlathImkeller2011">{{cite book|author1=Jochen Blath|author2=Peter Imkeller|author3=Sylvie Roelly|author3-link=Sylvie Roelly|title=Surveys in Stochastic Processes|url=https://books.google.com/books?id=CyK6KAjwdYkC|year=2011|publisher=European Mathematical Society|isbn=978-3-03719-072-2}}</ref><ref name="Talagrand2014">{{cite book|author=Michel Talagrand|title=Upper and Lower Bounds for Stochastic Processes: Modern Methods and Classical Problems|url=https://books.google.com/books?id=tfa5BAAAQBAJ&pg=PR4|year=2014|publisher=Springer Science & Business Media|isbn=978-3-642-54075-2|pages=4–}}</ref><ref name="Bressloff2014VII">{{cite book|author=[[Paul C. Bressloff]]|title=Stochastic Processes in Cell Biology|url=https://books.google.com/books?id=SwZYBAAAQBAJ&pg=PA1|year=2014|publisher=Springer|isbn=978-3-319-08488-6|pages=vii–ix}}</ref>