Editing Stochastic process (section)

==Introduction==
A stochastic or random process can be defined as a collection of random variables that is indexed by some mathematical set, meaning that each random variable of the stochastic process is uniquely associated with an element in the set.<ref name="Parzen1999"/><ref name="GikhmanSkorokhod1969page1"/> The set used to index the random variables is called the [[index set]]. Historically, the index set was some [[subset]] of the [[real line]], such as the [[natural numbers]], giving the index set the interpretation of time.<ref name="doob1953stochasticP46to47"/> Each random variable in the collection takes values from the same [[mathematical space]] known as the '''state space'''. This state space can be, for example, the integers, the real line or <math>n</math>-dimensional Euclidean space.<ref name="doob1953stochasticP46to47"/><ref name="GikhmanSkorokhod1969page1"/> An '''increment''' is the amount that a stochastic process changes between two index values, often interpreted as two points in time.<ref name="KarlinTaylor2012page27"/><ref name="Applebaum2004page1337"/> A stochastic process can have many [[Outcome (probability)|outcomes]], due to its randomness, and a single outcome of a stochastic process is called, among other names, a '''sample function''' or '''realization'''.<ref name="Lamperti1977page1"/><ref name="RogersWilliams2000page121b"/>

[[File:Wiener process 3d.png|thumb|right|A single computer-simulated '''sample function''' or '''realization''', among other terms, of a three-dimensional Wiener or Brownian motion process for time 0 ≤ t ≤ 2. The index set of this stochastic process is the non-negative numbers, while its state space is three-dimensional Euclidean space.]]

===Classifications===
A stochastic process can be classified in different ways, for example, by its state space, its index set, or the dependence among the random variables. One common way of classification is by the [[cardinality]] of the index set and the state space.<ref name="Florescu2014page294"/><ref name="KarlinTaylor2012page26">{{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|page=26}}</ref><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|pages=24, 25}}</ref>

When interpreted as time, if the index set of a stochastic process has a finite or countable number of elements, such as a finite set of numbers, the set of integers, or the natural numbers, then the stochastic process is said to be in '''[[discrete time]]'''.<ref name="Billingsley2008page482"/><ref name="Borovkov2013page527">{{cite book|author=Alexander A. Borovkov|title=Probability Theory|url=https://books.google.com/books?id=hRk_AAAAQBAJ|year=2013|publisher=Springer Science & Business Media|isbn=978-1-4471-5201-9|page=527}}</ref> If the index set is some interval of the real line, then time is said to be '''[[continuous time|continuous]]'''. The two types of stochastic processes are respectively referred to as '''discrete-time''' and '''[[continuous-time stochastic process]]es'''.<ref name="KarlinTaylor2012page27"/><ref name="Brémaud2014page120"/><ref name="Rosenthal2006page177">{{cite book|author=Jeffrey S Rosenthal|title=A First Look at Rigorous Probability Theory|url=https://books.google.com/books?id=am1IDQAAQBAJ|year=2006|publisher=World Scientific Publishing Co Inc|isbn=978-981-310-165-4|pages=177–178}}</ref> Discrete-time stochastic processes are considered easier to study because continuous-time processes require more advanced mathematical techniques and knowledge, particularly due to the index set being uncountable.<ref name="KloedenPlaten2013page63">{{cite book|author1=Peter E. Kloeden|author2=Eckhard Platen|title=Numerical Solution of Stochastic Differential Equations|url=https://books.google.com/books?id=r9r6CAAAQBAJ|year=2013|publisher=Springer Science & Business Media|isbn=978-3-662-12616-5|page=63}}</ref><ref name="Khoshnevisan2006page153">{{cite book|author1-link=Davar Khoshnevisan|author=Davar Khoshnevisan|title=Multiparameter Processes: An Introduction to Random Fields|url=https://books.google.com/books?id=XADpBwAAQBAJ|year=2006|publisher=Springer Science & Business Media|isbn=978-0-387-21631-7|pages=153–155}}</ref> If the index set is the integers, or some subset of them, then the stochastic process can also be called a '''random sequence'''.<ref name="Borovkov2013page527"/>

If the state space is the integers or natural numbers, then the stochastic process is called a '''discrete''' or '''integer-valued stochastic process'''. If the state space is the real line, then the stochastic process is referred to as a '''real-valued stochastic process''' or a '''process with continuous state space'''. If the state space is <math>n</math>-dimensional Euclidean space, then the stochastic process is called a <math>n</math>-'''dimensional vector process''' or <math>n</math>-'''vector process'''.<ref name="Florescu2014page294"/><ref name="KarlinTaylor2012page26"/>

===Etymology===
The word ''stochastic'' in [[English language|English]] was originally used as an adjective with the definition "pertaining to conjecturing", and stemming from a [[Greek language|Greek]] word meaning "to aim at a mark, guess", and the [[Oxford English Dictionary]] gives the year 1662 as its earliest occurrence.<ref name="OxfordStochastic">{{Cite OED|Stochastic}}</ref> 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".<ref name="Sheĭnin2006page5">{{cite book|author=O. B. Sheĭnin|title=Theory of probability and statistics as exemplified in short dictums|url=https://books.google.com/books?id=XqMZAQAAIAAJ|year=2006|publisher=NG Verlag|isbn=978-3-938417-40-9|page=5}}</ref> This phrase was used, with reference to Bernoulli, by [[Ladislaus Bortkiewicz]]<ref name="SheyninStrecker2011page136">{{cite book|author1=Oscar Sheynin|author2=Heinrich Strecker|title=Alexandr A. Chuprov: Life, Work, Correspondence|url=https://books.google.com/books?id=1EJZqFIGxBIC&pg=PA9|year=2011|publisher=V&R unipress GmbH|isbn=978-3-89971-812-6|page=136}}</ref> 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]].<ref name="OxfordStochastic"/> 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]],<ref name="Doob1934"/><ref name="Khintchine1934">{{cite journal|last1=Khintchine|first1=A.|title=Korrelationstheorie der stationeren stochastischen Prozesse|journal=Mathematische Annalen|volume=109|issue=1|year=1934|pages=604–615|issn=0025-5831|doi=10.1007/BF01449156|s2cid=122842868}}</ref> though the German term had been used earlier, for example, by Andrei Kolmogorov in 1931.<ref name="Kolmogoroff1931page1">{{cite journal|last1=Kolmogoroff|first1=A.|title=Über die analytischen Methoden in der Wahrscheinlichkeitsrechnung|journal=Mathematische Annalen|volume=104|issue=1|year=1931|page=1|issn=0025-5831|doi=10.1007/BF01457949|s2cid=119439925}}</ref>

According to the Oxford English Dictionary, early occurrences of the word ''random'' in English with its current meaning, which relates to chance or luck, date back to the 16th century, while earlier recorded usages started in the 14th century as a noun meaning "impetuosity, great speed, force, or violence (in riding, running, striking, etc.)". The word itself comes from a Middle French word meaning "speed, haste", and it is probably derived from a French verb meaning "to run" or "to gallop". The first written appearance of the term ''random process'' pre-dates ''stochastic process'', which the Oxford English Dictionary also gives as a synonym, and was used in an article by [[Francis Edgeworth]] published in 1888.<ref name="OxfordRandom">{{Cite OED|Random}}</ref>

===Terminology===
The definition of a stochastic process varies,<ref name="FristedtGray2013page580">{{cite book|author1=Bert E. Fristedt|author2=Lawrence F. Gray|title=A Modern Approach to Probability Theory|url=https://books.google.com/books?id=9xT3BwAAQBAJ&pg=PA716|year= 2013|publisher=Springer Science & Business Media|isbn=978-1-4899-2837-5|page=580}}</ref> but a stochastic process is traditionally defined as a collection of random variables indexed by some set.<ref name="RogersWilliams2000page121"/><ref name="Asmussen2003page408"/> The terms ''random process'' and ''stochastic process'' are considered synonyms and are used interchangeably, without the index set being precisely specified.<ref name="Kallenberg2002page24"/><ref name="ChaumontYor2012"/><ref name="AdlerTaylor2009page7"/><ref name="Stirzaker2005page45">{{cite book|author=David Stirzaker|title=Stochastic Processes and Models|url=https://books.google.com/books?id=0avUelS7e7cC|year=2005|publisher=Oxford University Press|isbn=978-0-19-856814-8|page=45}}</ref><ref name="Rosenblatt1962page91">{{cite book|author=Murray Rosenblatt|title=Random Processes|url=https://archive.org/details/randomprocesses00rose_0|url-access=registration|year=1962|publisher=Oxford University Press|page=[https://archive.org/details/randomprocesses00rose_0/page/91 91]}}</ref><ref name="Gubner2006page383">{{cite book|author=John A. Gubner|title=Probability and Random Processes for Electrical and Computer Engineers|url=https://books.google.com/books?id=pa20eZJe4LIC|year=2006|publisher=Cambridge University Press|isbn=978-1-139-45717-0|page=383}}</ref> Both "collection",<ref name="Lamperti1977page1"/><ref name="Stirzaker2005page45"/> or "family" are used<ref name="Parzen1999"/><ref name="Ito2006page13">{{cite book|author=Kiyosi Itō|title=Essentials of Stochastic Processes|url=https://books.google.com/books?id=pY5_DkvI-CcC&pg=PR4|year=2006|publisher=American Mathematical Soc.|isbn=978-0-8218-3898-3|page=13}}</ref> while instead of "index set", sometimes the terms "parameter set"<ref name="Lamperti1977page1"/> or "parameter space"<ref name="AdlerTaylor2009page7"/> are used.

The term ''random function'' is also used to refer to a stochastic or random process,<ref name="GikhmanSkorokhod1969page1"/><ref name="Loeve1978">{{cite book|author=M. Loève|title=Probability Theory II|url=https://books.google.com/books?id=1y229yBbULIC|year=1978|publisher=Springer Science & Business Media|isbn=978-0-387-90262-3|page=163}}</ref><ref name="Brémaud2014page133">{{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|page=133}}</ref> though sometimes it is only used when the stochastic process takes real values.<ref name="Lamperti1977page1"/><ref name="Ito2006page13"/> This term is also used when the index sets are mathematical spaces other than the real line,<ref name="GikhmanSkorokhod1969page1"/><ref name="GusakKukush2010page1">{{harvtxt|Gusak|Kukush|Kulik|Mishura|2010}}, p. 1</ref> while the terms ''stochastic process'' and ''random process'' are usually used when the index set is interpreted as time,<ref name="GikhmanSkorokhod1969page1"/><ref name="GusakKukush2010page1"/><ref name="Bass2011page1">{{cite book|author=Richard F. Bass|title=Stochastic Processes|url=https://books.google.com/books?id=Ll0T7PIkcKMC|year=2011|publisher=Cambridge University Press|isbn=978-1-139-50147-7|page=1}}</ref> and other terms are used such as ''random field'' when the index set is <math>n</math>-dimensional Euclidean space <math>\mathbb{R}^n</math> or a [[manifold]].<ref name="GikhmanSkorokhod1969page1"/><ref name="Lamperti1977page1"/><ref name="AdlerTaylor2009page7"/>

===Notation===
A stochastic process can be denoted, among other ways, by <math>\{X(t)\}_{t\in T} </math>,<ref name="Brémaud2014page120"/> <math>\{X_t\}_{t\in T} </math>,<ref name="Asmussen2003page408"/> <math>\{X_t\}</math><ref name="Lamperti1977page3">,{{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|page=3}}</ref> <math>\{X(t)\}</math> or simply as <math>X</math>. Some authors mistakenly write <math>X(t)</math> even though it is an [[abuse of notation#Function notation|abuse of function notation]].<ref name="Klebaner2005page55">{{cite book|author=Fima C. Klebaner|title=Introduction to Stochastic Calculus with Applications|url=https://books.google.com/books?id=JYzW0uqQxB0C|year=2005|publisher=Imperial College Press|isbn=978-1-86094-555-7|page=55}}</ref> For example, <math>X(t)</math> or <math>X_t</math> are used to refer to the random variable with the index <math>t</math>, and not the entire stochastic process.<ref name="Lamperti1977page3"/> If the index set is <math>T=[0,\infty)</math>, then one can write, for example, <math>(X_t , t \geq 0)</math> to denote the stochastic process.<ref name="ChaumontYor2012"/>