Editing Stochastic process (section)

==Examples==

===Bernoulli process===
{{Main|Bernoulli process}}

One of the simplest stochastic processes is the [[Bernoulli process]],<ref name="Florescu2014page293"/> which is a sequence of [[independent and identically distributed]] (iid) random variables, where each random variable takes either the value one or zero, say one with probability <math>p</math> and zero with probability <math>1-p</math>. This process can be linked to an idealisation of repeatedly flipping a coin, where the probability of obtaining a head is taken to be <math>p</math> and its value is one, while the value of a tail is zero.<ref name= "Florescu2014page301">{{cite book| first= Ionut |last= Florescu|title=Probability and Stochastic Processes|url=https://books.google.com/books?id=Z5xEBQAAQBAJ&pg=PR22|year=2014|publisher=John Wiley & Sons|isbn=978-1-118-59320-2|page=301}}</ref> In other words, a Bernoulli process is a sequence of iid Bernoulli random variables,<ref name="BertsekasTsitsiklis2002page273">{{cite book| first1= Dimitri P.| last1= Bertsekas| first2= John N. |last2= Tsitsiklis|title=Introduction to Probability |url= https://books.google.com/books?id=bcHaAAAAMAAJ|year=2002|publisher= Athena Scientific| isbn=978-1-886529-40-3|page=273}}</ref> where each idealised coin flip is an example of a [[Bernoulli trial]].<ref name="Ibe2013page11">{{cite book| first= Oliver C. |last= Ibe |title= Elements of Random Walk and Diffusion Processes|url=https://books.google.com/books?id=DUqaAAAAQBAJ&pg=PT10|year=2013|publisher=John Wiley & Sons|isbn=978-1-118-61793-9 |page= 11}}</ref>

===Random walk===
{{Main|Random walk}}

[[Random walks]] are stochastic processes that are usually defined as sums of [[iid]] random variables or random vectors in Euclidean space, so they are processes that change in discrete time.<ref name="Klenke2013page347">{{cite book|author=Achim Klenke|title=Probability Theory: A Comprehensive Course|url=https://books.google.com/books?id=aqURswEACAAJ|year=2013|publisher=Springer|isbn=978-1-4471-5362-7|pages=347}}</ref><ref name="LawlerLimic2010page1">{{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|page=1}}</ref><ref name="Kallenberg2002page136">{{cite book|author=Olav Kallenberg|title=Foundations of Modern Probability|url=https://books.google.com/books?id=L6fhXh13OyMC|date= 2002|publisher=Springer Science & Business Media|isbn=978-0-387-95313-7|page=136}}</ref><ref name="Florescu2014page383">{{cite book|author=Ionut Florescu|title=Probability and Stochastic Processes|url=https://books.google.com/books?id=Z5xEBQAAQBAJ&pg=PR22|year=2014|publisher=John Wiley & Sons|isbn=978-1-118-59320-2|page=383}}</ref><ref name="Durrett2010page277">{{cite book|author=Rick Durrett|title=Probability: Theory and Examples|url=https://books.google.com/books?id=evbGTPhuvSoC|year=2010|publisher=Cambridge University Press|isbn=978-1-139-49113-6|page=277}}</ref> But some also use the term to refer to processes that change in continuous time,<ref name="Weiss2006page1">{{cite book|last1=Weiss|first1=George H.|title=Encyclopedia of Statistical Sciences|chapter=Random Walks|year=2006|doi=10.1002/0471667196.ess2180.pub2|page=1|isbn=978-0471667193}}</ref> particularly the Wiener process used in financial models, which has led to some confusion, resulting in its criticism.<ref name="Spanos1999page454">{{cite book|author=Aris Spanos|title=Probability Theory and Statistical Inference: Econometric Modeling with Observational Data|url=https://books.google.com/books?id=G0_HxBubGAwC|year=1999|publisher=Cambridge University Press|isbn=978-0-521-42408-0|page=454}}</ref> There are various other types of random walks, defined so their state spaces can be other mathematical objects, such as lattices and groups, and in general they are highly studied and have many applications in different disciplines.<ref name="Weiss2006page1"/><ref name="Klebaner2005page81">{{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=81}}</ref>

A classic example of a random walk is known as the ''simple random walk'', which is a stochastic process in discrete time with the integers as the state space, and is based on a Bernoulli process, where each Bernoulli variable takes either the value positive one or negative one. In other words, the simple random walk takes place on the integers, and its value increases by one with probability, say, <math>p</math>, or decreases by one with probability <math>1-p</math>, so the index set of this random walk is the natural numbers, while its state space is the integers. If <math>p=0.5</math>, this random walk is called a symmetric random walk.<ref name="Gut2012page88">{{cite book|author=Allan Gut|title=Probability: A Graduate Course|url=https://books.google.com/books?id=XDFA-n_M5hMC|year=2012|publisher=Springer Science & Business Media|isbn=978-1-4614-4708-5|page=88}}</ref><ref name="GrimmettStirzaker2001page71">{{cite book|author1=Geoffrey Grimmett|author2=David Stirzaker|title=Probability and Random Processes|url=https://books.google.com/books?id=G3ig-0M4wSIC|year=2001|publisher=OUP Oxford|isbn=978-0-19-857222-0|page=71}}</ref>

===Wiener process===
{{Main|Wiener process}}

The Wiener process is a stochastic process with [[stationary increments|stationary]] and [[independent increments]] that are [[normally distributed]] based on the size of the increments.<ref name="RogersWilliams2000page1">{{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|page=1}}</ref><ref name="Klebaner2005page56">{{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=56}}</ref> The Wiener process is named after [[Norbert Wiener]], who proved its mathematical existence, but the process is also called the Brownian motion process or just Brownian motion due to its historical connection as a model for [[Brownian movement]] in liquids.<ref name="Brush1968page1">{{cite journal|last1=Brush|first1=Stephen G.|title=A history of random processes|journal=Archive for History of Exact Sciences|volume=5|issue=1|year=1968|pages=1–2|issn=0003-9519|doi=10.1007/BF00328110|s2cid=117623580}}</ref><ref name="Applebaum2004page1338">{{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=1338}}</ref><ref name="GikhmanSkorokhod1969page21">{{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&pg=PR2|year=1969|publisher=Courier Corporation|isbn=978-0-486-69387-3|page=21}}</ref>

[[File:DriftedWienerProcess1D.svg|thumb|left|Realizations of Wiener processes (or Brownian motion processes) with drift ({{color|blue|blue}}) and without drift ({{color|red|red}})]]

Playing a central role in the theory of probability, the Wiener process is often considered the most important and studied stochastic process, with connections to other stochastic processes.<ref name="doob1953stochasticP46to47"/><ref name="RogersWilliams2000page1"/><ref name="Steele2012page29">{{cite book|author=J. Michael Steele|title=Stochastic Calculus and Financial Applications|url=https://books.google.com/books?id=fsgkBAAAQBAJ&pg=PR4|year=2012|publisher=Springer Science & Business Media|isbn=978-1-4684-9305-4|page=29}}</ref><ref name="Florescu2014page471">{{cite book|author=Ionut Florescu|title=Probability and Stochastic Processes|url=https://books.google.com/books?id=Z5xEBQAAQBAJ&pg=PR22|year=2014|publisher=John Wiley & Sons|isbn=978-1-118-59320-2|page=471}}</ref><ref name="KarlinTaylor2012page21">{{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|pages=21, 22}}</ref><ref name="KaratzasShreve2014pageVIII">{{cite book|author1=Ioannis Karatzas|author2=Steven Shreve|title=Brownian Motion and Stochastic Calculus|url=https://books.google.com/books?id=w0SgBQAAQBAJ&pg=PT5|year=1991|publisher=Springer|isbn=978-1-4612-0949-2|page=VIII}}</ref><ref name="RevuzYor2013pageIX">{{cite book|author1=Daniel Revuz|author2=Marc Yor|title=Continuous Martingales and Brownian Motion|url=https://books.google.com/books?id=OYbnCAAAQBAJ|year=2013|publisher=Springer Science & Business Media|isbn=978-3-662-06400-9|page=IX|author1-link=Daniel Revuz}}</ref> Its index set and state space are the non-negative numbers and real numbers, respectively, so it has both continuous index set and states space.<ref name="Rosenthal2006page186">{{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|page=186}}</ref> But the process can be defined more generally so its state space can be <math>n</math>-dimensional Euclidean space.<ref name="Klebaner2005page81"/><ref name="KarlinTaylor2012page21"/><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=33}}</ref> If the [[mean]] of any increment is zero, then the resulting Wiener or Brownian motion process is said to have zero drift. If the mean of the increment for any two points in time is equal to the time difference multiplied by some constant <math> \mu</math>, which is a real number, then the resulting stochastic process is said to have drift <math> \mu</math>.<ref name="Steele2012page118">{{cite book|author=J. Michael Steele|title=Stochastic Calculus and Financial Applications|url=https://books.google.com/books?id=fsgkBAAAQBAJ&pg=PR4|year=2012|publisher=Springer Science & Business Media|isbn=978-1-4684-9305-4|page=118}}</ref><ref name="MörtersPeres2010page1"/><ref name="KaratzasShreve2014page78">{{cite book|author1=Ioannis Karatzas|author2=Steven Shreve|title=Brownian Motion and Stochastic Calculus|url=https://books.google.com/books?id=w0SgBQAAQBAJ&pg=PT5|year=1991|publisher=Springer|isbn=978-1-4612-0949-2|page=78}}</ref>

[[Almost surely]], a sample path of a Wiener process is continuous everywhere but [[nowhere differentiable function|nowhere differentiable]]. It can be considered as a continuous version of the simple random walk.<ref name="Applebaum2004page1337">{{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|page=1337}}</ref><ref name="MörtersPeres2010page1">{{cite book|author1=Peter Mörters|author2=Yuval Peres|title=Brownian Motion|url=https://books.google.com/books?id=e-TbA-dSrzYC|year=2010|publisher=Cambridge University Press|isbn=978-1-139-48657-6|pages=1, 3}}</ref> The process arises as the mathematical limit of other stochastic processes such as certain random walks rescaled,<ref name="KaratzasShreve2014page61">{{cite book|author1=Ioannis Karatzas|author2=Steven Shreve|title=Brownian Motion and Stochastic Calculus|url=https://books.google.com/books?id=w0SgBQAAQBAJ&pg=PT5|year=1991|publisher=Springer|isbn=978-1-4612-0949-2|page=61}}</ref><ref name="Shreve2004page93">{{cite book|author=Steven E. 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|page=93}}</ref> which is the subject of [[Donsker's theorem]] or invariance principle, also known as the functional central limit theorem.<ref name="Kallenberg2002page225and260">{{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=225, 260}}</ref><ref name="KaratzasShreve2014page70">{{cite book|author1=Ioannis Karatzas|author2=Steven Shreve|title=Brownian Motion and Stochastic Calculus|url=https://books.google.com/books?id=w0SgBQAAQBAJ&pg=PT5|year=1991|publisher=Springer|isbn=978-1-4612-0949-2|page=70}}</ref><ref name="MörtersPeres2010page131">{{cite book|author1=Peter Mörters|author2=Yuval Peres|title=Brownian Motion|url=https://books.google.com/books?id=e-TbA-dSrzYC|year=2010|publisher=Cambridge University Press|isbn=978-1-139-48657-6|page=131}}</ref>

The Wiener process is a member of some important families of stochastic processes, including Markov processes, Lévy processes and Gaussian processes.<ref name="RogersWilliams2000page1"/><ref name="Applebaum2004page1337"/> The process also has many applications and is the main stochastic process used in stochastic calculus.<ref name="Klebaner2005">{{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}}</ref><ref name="KaratzasShreve2014page">{{cite book|author1=Ioannis Karatzas|author2=Steven Shreve|title=Brownian Motion and Stochastic Calculus|url=https://books.google.com/books?id=w0SgBQAAQBAJ&pg=PT5|year=1991|publisher=Springer|isbn=978-1-4612-0949-2}}</ref> It plays a central role in quantitative finance,<ref name="Applebaum2004page1341">{{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|page=1341}}</ref><ref name="KarlinTaylor2012page340">{{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=340}}</ref> where it is used, for example, in the Black–Scholes–Merton model.<ref name="Klebaner2005page124">{{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=124}}</ref> The process is also used in different fields, including the majority of natural sciences as well as some branches of social sciences, as a mathematical model for various random phenomena.<ref name="Steele2012page29"/><ref name="KaratzasShreve2014page47">{{cite book|author1=Ioannis Karatzas|author2=Steven Shreve|title=Brownian Motion and Stochastic Calculus|url=https://books.google.com/books?id=w0SgBQAAQBAJ&pg=PT5|year=1991|publisher=Springer|isbn=978-1-4612-0949-2|page=47}}</ref><ref name="Wiersema2008page2">{{cite book|author=Ubbo F. Wiersema|title=Brownian Motion Calculus|url=https://books.google.com/books?id=0h-n0WWuD9cC|year=2008|publisher=John Wiley & Sons|isbn=978-0-470-02171-2|page=2}}</ref>

===Poisson process===
{{Main|Poisson process}}

The Poisson process is a stochastic process that has different forms and definitions.<ref name="Tijms2003page1">{{cite book|author=Henk C. Tijms|title=A First Course in Stochastic Models|url=https://books.google.com/books?id=eBeNngEACAAJ|year=2003|publisher=Wiley|isbn=978-0-471-49881-0|pages=1, 2}}</ref><ref name="DaleyVere-Jones2006chap2">{{cite book|author1=D.J. Daley|author2=D. Vere-Jones|title=An Introduction to the Theory of Point Processes: Volume I: Elementary Theory and Methods|url=https://books.google.com/books?id=6Sv4BwAAQBAJ|year=2006|publisher=Springer Science & Business Media|isbn=978-0-387-21564-8|pages=19–36}}</ref> It can be defined as a counting process, which is a stochastic process that represents the random number of points or events up to some time. The number of points of the process that are located in the interval from zero to some given time is a Poisson random variable that depends on that time and some parameter. This process has the natural numbers as its state space and the non-negative numbers as its index set. This process is also called the Poisson counting process, since it can be interpreted as an example of a counting process.<ref name="Tijms2003page1"/>

If a Poisson process is defined with a single positive constant, then the process is called a homogeneous Poisson process.<ref name="Tijms2003page1"/><ref name="PinskyKarlin2011">{{cite book|author1=Mark A. Pinsky|author2=Samuel Karlin|title=An Introduction to Stochastic Modeling|url=https://books.google.com/books?id=PqUmjp7k1kEC|year=2011|publisher=Academic Press|isbn=978-0-12-381416-6|page=241}}</ref> The homogeneous Poisson process is a member of important classes of stochastic processes such as Markov processes and Lévy processes.<ref name="Applebaum2004page1337"/>

The homogeneous Poisson process can be defined and generalized in different ways. It can be defined such that its index set is the real line, and this stochastic process is also called the stationary Poisson process.<ref name="Kingman1992page38">{{cite book|author=J. F. C. Kingman|title=Poisson Processes|url=https://books.google.com/books?id=VEiM-OtwDHkC|year=1992|publisher=Clarendon Press|isbn=978-0-19-159124-2|page=38}}</ref><ref name="DaleyVere-Jones2006page19">{{cite book|author1=D.J. Daley|author2=D. Vere-Jones|title=An Introduction to the Theory of Point Processes: Volume I: Elementary Theory and Methods|url=https://books.google.com/books?id=6Sv4BwAAQBAJ|year=2006|publisher=Springer Science & Business Media|isbn=978-0-387-21564-8|page=19}}</ref> If the parameter constant of the Poisson process is replaced with some non-negative integrable function of <math>t</math>, the resulting process is called an inhomogeneous or nonhomogeneous Poisson process, where the average density of points of the process is no longer constant.<ref name="Kingman1992page22">{{cite book|author=J. F. C. Kingman|title=Poisson Processes|url=https://books.google.com/books?id=VEiM-OtwDHkC|year=1992|publisher=Clarendon Press|isbn=978-0-19-159124-2|page=22}}</ref> Serving as a fundamental process in queueing theory, the Poisson process is an important process for mathematical models, where it finds applications for models of events randomly occurring in certain time windows.<ref name="KarlinTaylor2012page118">{{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|pages=118, 119}}</ref><ref name="Kleinrock1976page61">{{cite book|author=Leonard Kleinrock|title=Queueing Systems: Theory|url=https://archive.org/details/queueingsystems00klei|url-access=registration|year=1976|publisher=Wiley|isbn=978-0-471-49110-1|page=[https://archive.org/details/queueingsystems00klei/page/61 61]}}</ref>

Defined on the real line, the Poisson process can be interpreted as a stochastic process,<ref name="Applebaum2004page1337"/><ref name="Rosenblatt1962page94">{{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/94 94]}}</ref> among other random objects.<ref name="Haenggi2013page10and18">{{cite book|author=Martin Haenggi|title=Stochastic Geometry for Wireless Networks|url=https://books.google.com/books?id=CLtDhblwWEgC|year=2013|publisher=Cambridge University Press|isbn=978-1-107-01469-5|pages=10, 18}}</ref><ref name="ChiuStoyan2013page41and108">{{cite book|author1=Sung Nok Chiu|author2=Dietrich Stoyan|author3=Wilfrid S. Kendall|author4=Joseph Mecke|title=Stochastic Geometry and Its Applications|url=https://books.google.com/books?id=825NfM6Nc-EC|year=2013|publisher=John Wiley & Sons|isbn=978-1-118-65825-3|pages=41, 108}}</ref> But then it can be defined on the <math>n</math>-dimensional Euclidean space or other mathematical spaces,<ref name="Kingman1992page11">{{cite book|author=J. F. C. Kingman|title=Poisson Processes|url=https://books.google.com/books?id=VEiM-OtwDHkC|year=1992|publisher=Clarendon Press|isbn=978-0-19-159124-2|page=11}}</ref> where it is often interpreted as a random set or a random counting measure, instead of a stochastic process.<ref name="Haenggi2013page10and18"/><ref name="ChiuStoyan2013page41and108"/> In this setting, the Poisson process, also called the Poisson point process, is one of the most important objects in probability theory, both for applications and theoretical reasons.<ref name="Stirzaker2000"/><ref name="Streit2010page1">{{cite book|author=Roy L. Streit|title=Poisson Point Processes: Imaging, Tracking, and Sensing|url=https://books.google.com/books?id=KAWmFYUJ5zsC&pg=PA11|year=2010|publisher=Springer Science & Business Media|isbn=978-1-4419-6923-1|page=1}}</ref> But it has been remarked that the Poisson process does not receive as much attention as it should, partly due to it often being considered just on the real line, and not on other mathematical spaces.<ref name="Streit2010page1"/><ref name="Kingman1992pagev">{{cite book|author=J. F. C. Kingman|title=Poisson Processes|url=https://books.google.com/books?id=VEiM-OtwDHkC|year=1992|publisher=Clarendon Press|isbn=978-0-19-159124-2|page=v}}</ref>