Editing Stochastic process (section)

===Discoveries of specific stochastic processes===
Although Khinchin gave mathematical definitions of stochastic processes in the 1930s,<ref name="Doob1934"/><ref name="Vere-Jones2006page4"/> specific stochastic processes had already been discovered in different settings, such as the Brownian motion process and the Poisson process.<ref name="JarrowProtter2004"/><ref name="GuttorpThorarinsdottir2012"/> Some families of stochastic processes such as point processes or renewal processes have long and complex histories, stretching back centuries.<ref name="DaleyVere-Jones2006chap1">{{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=1–4}}</ref>

====Bernoulli process====
The Bernoulli process, which can serve as a mathematical model for flipping a biased coin, is possibly the first stochastic process to have been studied.<ref name="Florescu2014page301"/> The process is a sequence of independent Bernoulli trials,<ref name="BertsekasTsitsiklis2002page273"/> which are named after [[Jacob Bernoulli]] who used them to study games of chance, including probability problems proposed and studied earlier by Christiaan Huygens.<ref name="Hald2005page226">{{cite book|author=Anders Hald|title=A History of Probability and Statistics and Their Applications before 1750|url=https://books.google.com/books?id=pOQy6-qnVx8C|year=2005|publisher=John Wiley & Sons|isbn=978-0-471-72517-6|page=226}}</ref> Bernoulli's work, including the Bernoulli process, were published in his book ''Ars Conjectandi'' in 1713.<ref name="Lebowitz1984">{{cite book|author=Joel Louis Lebowitz|title=Nonequilibrium phenomena II: from stochastics to hydrodynamics|url=https://books.google.com/books?id=E8IRAQAAIAAJ|year=1984|publisher=North-Holland Pub.|isbn=978-0-444-86806-0|pages=8–10}}</ref>

====Random walks====
In 1905, [[Karl Pearson]] coined the term ''random walk'' while posing a problem describing a random walk on the plane, which was motivated by an application in biology, but such problems involving random walks had already been studied in other fields. Certain gambling problems that were studied centuries earlier can be considered as problems involving random walks.<ref name="Weiss2006page1"/><ref name="Lebowitz1984"/> For example, the problem known as the ''Gambler's ruin'' is based on a simple random walk,<ref name="KarlinTaylor2012page49"/><ref name="Florescu2014page374">{{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|pages=374}}</ref> and is an example of a random walk with absorbing barriers.<ref name="Seneta2006page1"/><ref name="Ibe2013page5">{{cite book|author=Oliver C. 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=5}}</ref> Pascal, Fermat and Huyens all gave numerical solutions to this problem without detailing their methods,<ref name="Hald2005page63">{{cite book|author=Anders Hald|title=A History of Probability and Statistics and Their Applications before 1750|url=https://books.google.com/books?id=pOQy6-qnVx8C|year=2005|publisher=John Wiley & Sons|isbn=978-0-471-72517-6|page=63}}</ref> and then more detailed solutions were presented by Jakob Bernoulli and [[Abraham de Moivre]].<ref name="Hald2005page202">{{cite book|author=Anders Hald|title=A History of Probability and Statistics and Their Applications before 1750|url=https://books.google.com/books?id=pOQy6-qnVx8C|year=2005|publisher=John Wiley & Sons|isbn=978-0-471-72517-6|page=202}}</ref>

For random walks in <math>n</math>-dimensional integer [[Lattice (group)|lattices]], [[George Pólya]] published, in 1919 and 1921, work where he studied the probability of a symmetric random walk returning to a previous position in the lattice. Pólya showed that a symmetric random walk, which has an equal probability to advance in any direction in the lattice, will return to a previous position in the lattice an infinite number of times with probability one in one and two dimensions, but with probability zero in three or higher dimensions.<ref name="Florescu2014page385">{{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|pages=385}}</ref><ref name="Hughes1995page111">{{cite book|author=Barry D. Hughes|title=Random Walks and Random Environments: Random walks|url=https://books.google.com/books?id=QhOen_t0LeQC|year=1995|publisher=Clarendon Press|isbn=978-0-19-853788-5|page=111}}</ref>

====Wiener process====
The [[Wiener process]] or Brownian motion process has its origins in different fields including statistics, finance and physics.<ref name="JarrowProtter2004"/> In 1880, Danish astronomer  [[Thorvald Thiele]] wrote a paper on the method of least squares, where he used the process to study the errors of a model in time-series analysis.<ref name="Thiele1880">{{cite journal|last1=Thiele|first1=Thorwald N.|title=Om Anvendelse af mindste Kvadraterbs Methode i nogle Tilfælde, hvoren Komplikation af visse Slags uensartede tilfældige Fejlkilder giver Fejleneen "systematisk" Karakter|journal=Kongelige Danske Videnskabernes Selskabs Skrifter |volume=Series 5|issue=12|year=1880|pages=381–408|url=https://biodiversitylibrary.org/page/43213604}}</ref><ref name="Hald1981page1and18">{{cite journal|last1=Hald|first1=Anders |title=T. N. Thiele's Contributions to Statistics|journal=International Statistical Review / Revue Internationale de Statistique|volume=49|issue=1|year=1981|pages=1–20|issn=0306-7734|doi=10.2307/1403034|jstor=1403034}}</ref><ref name="Lauritzen1981page319">{{cite journal|last1=Lauritzen|first1=Steffen L.|title=Time Series Analysis in 1880: A Discussion of Contributions Made by T.N. Thiele|journal=International Statistical Review / Revue Internationale de Statistique|volume=49|issue=3|year=1981|pages=319–320|issn=0306-7734|doi=10.2307/1402616|jstor=1402616}}</ref> The work is now considered as an early discovery of the statistical method known as [[Kalman filtering]], but the work was largely overlooked. It is thought that the ideas in Thiele's paper were too advanced to have been understood by the broader mathematical and statistical community at the time.<ref name="Lauritzen1981page319"/>

[[File:Wiener Zurich1932.tif|thumb|200px|[[Norbert Wiener]] gave the first mathematical proof of the existence of the Wiener process. This mathematical object had appeared previously in the work of [[Thorvald Thiele]], [[Louis Bachelier]], and [[Albert Einstein]].<ref name="JarrowProtter2004"/>]]

The French mathematician [[Louis Bachelier]] used a Wiener process in his 1900 thesis<ref name=Bachelier1900a>{{cite journal |last=Bachelier |first=Luis |year=1900 |title=Théorie de la spéculation |journal=[[Ann. Sci. Éc. Norm. Supér.]] |volume=Serie 3;17 |pages=21–89 |url=http://archive.numdam.org/article/ASENS_1900_3_17__21_0.pdf |archive-url=https://web.archive.org/web/20110605013545/http://archive.numdam.org/article/ASENS_1900_3_17__21_0.pdf |archive-date=2011-06-05 |url-status=live |doi=10.24033/asens.476 |doi-access=free }}</ref><ref name=Bachelier1900b>{{cite journal |last=Bachelier |first=Luis |year=1900 |title=The Theory of Speculation |journal=Ann. Sci. Éc. Norm. Supér. |volume=Serie 3;17 |pages=21–89 (Engl. translation by David R. May, 2011) |url=https://drive.google.com/file/d/0B5LLDy7-d3SKNGI0M2E0NGItYzFlMS00NGU2LWE2ZDAtODc3MDY3MzdiNmY0/view |doi=10.24033/asens.476 |doi-access=free }}</ref> in order to model price changes on the [[Paris Bourse]], a [[stock exchange]],<ref name="CourtaultKabanov2000">{{cite journal|last1=Courtault|first1=Jean-Michel|last2=Kabanov|first2=Yuri|last3=Bru|first3=Bernard|last4=Crepel|first4=Pierre|last5=Lebon|first5=Isabelle|last6=Le Marchand|first6=Arnaud|title=Louis Bachelier on the Centenary of Theorie de la Speculation|journal=Mathematical Finance|volume=10|issue=3|year=2000|pages=339–353|issn=0960-1627|doi=10.1111/1467-9965.00098|s2cid=14422885 |url=https://halshs.archives-ouvertes.fr/halshs-00447592/file/BACHEL2.PDF |archive-url=https://web.archive.org/web/20180721214136/https://halshs.archives-ouvertes.fr/halshs-00447592/file/BACHEL2.PDF |archive-date=2018-07-21 |url-status=live}}</ref> without knowing the work of Thiele.<ref name="JarrowProtter2004"/> It has been speculated that Bachelier drew ideas from the random walk model of [[Jules Regnault]], but Bachelier did not cite him,<ref name="Jovanovic2012">{{cite journal|last1=Jovanovic|first1=Franck|title=Bachelier: Not the forgotten forerunner he has been depicted as. An analysis of the dissemination of Louis Bachelier's work in economics|journal=The European Journal of the History of Economic Thought|volume=19|issue=3|year=2012|pages=431–451|issn=0967-2567|doi=10.1080/09672567.2010.540343|s2cid=154003579|url=http://r-libre.teluq.ca/1168/1/dissemination%20of%20Louis%20Bachelier_EJHET_R2.pdf |archive-url=https://web.archive.org/web/20180721111017/http://r-libre.teluq.ca/1168/1/dissemination%20of%20Louis%20Bachelier_EJHET_R2.pdf |archive-date=2018-07-21 |url-status=live}}</ref> and Bachelier's thesis is now considered pioneering in the field of financial mathematics.<ref name="CourtaultKabanov2000"/><ref name="Jovanovic2012"/>

It is commonly thought that Bachelier's work gained little attention and was forgotten for decades until it was rediscovered in the 1950s by the [[Leonard Savage]], and then become more popular after Bachelier's thesis was translated into English in 1964. But the work was never forgotten in the mathematical community, as Bachelier published a book in 1912 detailing his ideas,<ref name="Jovanovic2012"/> which was cited by mathematicians including Doob, Feller<ref name="Jovanovic2012"/> and Kolmogorov.<ref name="JarrowProtter2004"/> The book continued to be cited, but then starting in the 1960s, the original thesis by Bachelier began to be cited more than his book when economists started citing Bachelier's work.<ref name="Jovanovic2012"/>

In 1905, [[Albert Einstein]] published a paper where he studied the physical observation of Brownian motion or movement to explain the seemingly random movements of particles in liquids by using ideas from the [[kinetic theory of gases]]. Einstein derived a [[differential equation]], known as a [[diffusion equation]], for describing the probability of finding a particle in a certain region of space. Shortly after Einstein's first paper on Brownian movement, [[Marian Smoluchowski]] published work where he cited Einstein, but wrote that he had independently derived the equivalent results by using a different method.<ref name="Brush1968page25">{{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|page=25|issn=0003-9519|doi=10.1007/BF00328110|s2cid=117623580}}</ref>

Einstein's work, as well as experimental results obtained by [[Jean Perrin]], later inspired Norbert Wiener in the 1920s<ref name="Brush1968page30">{{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=1–36|issn=0003-9519|doi=10.1007/BF00328110|s2cid=117623580}}</ref> to use a type of measure theory, developed by [[Percy Daniell]], and Fourier analysis to prove the existence of the Wiener process as a mathematical object.<ref name="JarrowProtter2004"/>

====Poisson process====
The Poisson process is named after [[Siméon Poisson]], due to its definition involving the [[Poisson distribution]], but Poisson never studied the process.<ref name="Stirzaker2000"/><ref name="DaleyVere-Jones2006page8">{{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=8–9}}</ref> There are a number of claims for early uses or discoveries of the Poisson
process.<ref name="Stirzaker2000"/><ref name="GuttorpThorarinsdottir2012"/>
At the beginning of the 20th century, the Poisson process would arise independently in different situations.<ref name="Stirzaker2000"/><ref name="GuttorpThorarinsdottir2012"/>
In Sweden 1903, [[Filip Lundberg]] published a [[thesis]] containing work, now considered fundamental and pioneering, where he proposed to model insurance claims with a homogeneous Poisson process.<ref name="EmbrechtsFrey2001page367">{{cite book|last1=Embrechts|first1=Paul|title=Stochastic Processes: Theory and Methods|last2=Frey|first2=Rüdiger|last3=Furrer|first3=Hansjörg|chapter=Stochastic processes in insurance and finance|volume=19|year=2001|page=367|issn=0169-7161|doi=10.1016/S0169-7161(01)19014-0|series=Handbook of Statistics|isbn=978-0444500144}}</ref><ref name="Cramér1969">{{cite journal|last1=Cramér|first1=Harald|title=Historical review of Filip Lundberg's works on risk theory|journal=Scandinavian Actuarial Journal|volume=1969|issue=sup3|year=1969|pages=6–12|issn=0346-1238|doi=10.1080/03461238.1969.10404602}}</ref>

Another discovery occurred in [[Denmark]] in 1909 when [[A.K. Erlang]] derived the Poisson distribution when developing a mathematical model for the number of incoming phone calls in a finite time interval. Erlang was not at the time aware of Poisson's earlier work and assumed that the number phone calls arriving in each interval of time were independent to each other. He then found the limiting case, which is effectively recasting the Poisson distribution as a limit of the binomial distribution.<ref name="Stirzaker2000"/>

In 1910, [[Ernest Rutherford]] and [[Hans Geiger]] published experimental results on counting alpha particles. Motivated by their work, [[Harry Bateman]] studied the counting problem and derived Poisson probabilities as a solution to a family of differential equations, resulting in the independent discovery of the Poisson process.<ref name="Stirzaker2000"/> After this time there were many studies and applications of the Poisson process, but its early history is complicated, which has been explained by the various applications of the process in numerous fields by biologists, ecologists, engineers and various physical scientists.<ref name="Stirzaker2000"/>

====Markov processes====
Markov processes and Markov chains are named after [[Andrey Markov]] who studied Markov chains in the early 20th century. Markov was interested in studying an extension of independent random sequences. In his first paper on Markov chains, published in 1906, Markov showed that under certain conditions the average outcomes of the Markov chain would converge to a fixed vector of values, so proving a [[weak law of large numbers]] without the independence assumption,<ref name="GrinsteadSnell1997page464">{{cite book|author1=Charles Miller Grinstead|author2=James Laurie Snell|title=Introduction to Probability|url=https://archive.org/details/flooved3489|year=1997|publisher=American Mathematical Soc.|isbn=978-0-8218-0749-1|pages=[https://archive.org/details/flooved3489/page/n473 464]–466}}</ref><ref name="Bremaud2013pageIX">{{cite book|author=Pierre Bremaud|title=Markov Chains: Gibbs Fields, Monte Carlo Simulation, and Queues|url=https://books.google.com/books?id=jrPVBwAAQBAJ|year=2013|publisher=Springer Science & Business Media|isbn=978-1-4757-3124-8|page=ix}}</ref><ref name="Hayes2013">{{cite journal|last1=Hayes|first1=Brian|title=First links in the Markov chain|journal=American Scientist|volume=101|issue=2|year=2013|pages=92–96|doi=10.1511/2013.101.92}}</ref> which had been commonly regarded as a requirement for such mathematical laws to hold.<ref name="Hayes2013"/> Markov later used Markov chains to study the distribution of vowels in [[Eugene Onegin]], written by [[Alexander Pushkin]], and proved a [[central limit theorem]] for such chains.

In 1912, Poincaré studied Markov chains on [[finite group]]s with an aim to study card shuffling. Other early uses of Markov chains include a diffusion model, introduced by [[Paul Ehrenfest|Paul]] and [[Tatyana Ehrenfest]] in 1907, and a branching process, introduced by [[Francis Galton]] and [[Henry William Watson]] in 1873, preceding the work of Markov.<ref name="GrinsteadSnell1997page464"/><ref name="Bremaud2013pageIX"/> After the work of Galton and Watson, it was later revealed that their branching process had been independently discovered and studied around three decades earlier by [[Irénée-Jules Bienaymé]].<ref name="Seneta1998">{{cite journal|last1=Seneta|first1=E.|title=I.J. Bienaymé [1796-1878]: Criticality, Inequality, and Internationalization|journal=International Statistical Review / Revue Internationale de Statistique|volume=66|issue=3|year=1998|pages=291–292|issn=0306-7734|doi=10.2307/1403518|jstor=1403518}}</ref> Starting in 1928, [[Maurice Fréchet]] became interested in Markov chains, eventually resulting in him publishing in 1938 a detailed study on Markov chains.<ref name="GrinsteadSnell1997page464"/><ref name="BruHertz2001">{{cite book|last1=Bru|first1=B.|title=Statisticians of the Centuries|last2=Hertz|first2=S.|chapter=Maurice Fréchet|year=2001|pages=331–334|doi=10.1007/978-1-4613-0179-0_71|isbn=978-0-387-95283-3}}</ref>

[[Andrei Kolmogorov]] developed in a 1931 paper a large part of the early theory of continuous-time Markov processes.<ref name="Cramer1976"/><ref name="KendallBatchelor1990page33"/> Kolmogorov was partly inspired by Louis Bachelier's 1900 work on fluctuations in the stock market as well as [[Norbert Wiener]]'s work on Einstein's model of Brownian movement.<ref name="KendallBatchelor1990page33"/><ref name="BarbutLocker2016page5">{{cite book|author1=Marc Barbut|author2=Bernard Locker|author3=Laurent Mazliak|title=Paul Lévy and Maurice Fréchet: 50 Years of Correspondence in 107 Letters|url=https://books.google.com/books?id=lSz_vQAACAAJ|date= 2016|publisher=Springer London|isbn=978-1-4471-7262-8|page=5}}</ref> He introduced and studied a particular set of Markov processes known as diffusion processes, where he derived a set of differential equations describing the processes.<ref name="KendallBatchelor1990page33"/><ref name="Skorokhod2005page146">{{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=146}}</ref> Independent of Kolmogorov's work, [[Sydney Chapman (mathematician)|Sydney Chapman]] derived in a 1928 paper an equation, now called the [[Chapman–Kolmogorov equation]], in a less mathematically rigorous way than Kolmogorov, while studying Brownian movement.<ref name="Bernstein2005">{{cite journal|last1=Bernstein|first1=Jeremy|title=Bachelier|journal=American Journal of Physics|volume=73|issue=5|year=2005|pages=398–396|issn=0002-9505|doi=10.1119/1.1848117|bibcode=2005AmJPh..73..395B}}</ref> The differential equations are now called the Kolmogorov equations<ref name="Anderson2012pageVII">{{cite book|author=William J. Anderson|title=Continuous-Time Markov Chains: An Applications-Oriented Approach|url=https://books.google.com/books?id=YpHfBwAAQBAJ&pg=PR8|year=2012|publisher=Springer Science & Business Media|isbn=978-1-4612-3038-0|page=vii}}</ref> or the Kolmogorov–Chapman equations.<ref name="KendallBatchelor1990page57">{{cite journal|last1=Kendall|first1=D. G.|last2=Batchelor|first2=G. K.|last3=Bingham|first3=N. H.|last4=Hayman|first4=W. K.|last5=Hyland|first5=J. M. E.|last6=Lorentz|first6=G. G.|last7=Moffatt|first7=H. K.|last8=Parry|first8=W.|last9=Razborov|first9=A. A.|last10=Robinson|first10=C. A.|last11=Whittle|first11=P.|title=Andrei Nikolaevich Kolmogorov (1903–1987)|journal=Bulletin of the London Mathematical Society|volume=22|issue=1|year=1990|page=57|issn=0024-6093|doi=10.1112/blms/22.1.31}}</ref> Other mathematicians who contributed significantly to the foundations of Markov processes include William Feller, starting in the 1930s, and then later Eugene Dynkin, starting in the 1950s.<ref name="Cramer1976"/>

====Lévy processes====
Lévy processes such as the Wiener process and the Poisson process (on the real line) are named after Paul Lévy who started studying them in the 1930s,<ref name="Applebaum2004page1336"/> but they have connections to [[infinitely divisible distribution]]s going back to the 1920s.<ref name="Bertoin1998pageVIII"/> In a 1932 paper, Kolmogorov derived a [[Characteristic function (probability theory)|characteristic function]] for random variables associated with Lévy processes. This result was later derived under more general conditions by Lévy in 1934, and then Khinchin independently gave an alternative form for this characteristic function in 1937.<ref name="Cramer1976"/><ref name="ApplebaumBook2004page67">{{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|page=67}}</ref> In addition to Lévy, Khinchin and Kolomogrov, early fundamental contributions to the theory of Lévy processes were made by [[Bruno de Finetti]] and [[Kiyosi Itô]].<ref name="Bertoin1998pageVIII"/>