Editing Stochastic process (section)

====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"/>