Editing Stochastic (section)

==Mathematics==
In the early 1930s, Aleksandr Khinchin gave the first mathematical definition of a stochastic process as a family of random variables indexed by the real line.<ref name="Vere-Jones2006page4">{{cite book|last1=Vere-Jones|first1=David|title=Encyclopedia of Statistical Sciences|chapter=Khinchin, Aleksandr Yakovlevich|page=4|year=2006|doi=10.1002/0471667196.ess6027.pub2|isbn=0471667196}}</ref><ref name="Doob1934"/>{{efn|Doob, when citing Khinchin, uses the term 'chance variable', which used to be an alternative term for 'random variable'.<ref name="Snell2005">{{cite journal|last1=Snell|first1=J. Laurie|title=Obituary: Joseph Leonard Doob|journal=Journal of Applied Probability|volume=42|issue=1|year=2005|page=251|issn=0021-9002|doi=10.1239/jap/1110381384|doi-access=free}}</ref> }} Further fundamental work on probability theory and stochastic processes was done by Khinchin as well as other mathematicians such as [[Andrey Kolmogorov]], [[Joseph Doob]], [[William Feller]], [[Maurice Fréchet]], [[Paul Lévy (mathematician)|Paul Lévy]], [[Wolfgang Doeblin]], and [[Harald Cramér]].<ref name="Bingham2000">{{cite journal|last1=Bingham|first1=N.|title=Studies in the history of probability and statistics XLVI. Measure into probability: from Lebesgue to Kolmogorov|journal=Biometrika|volume=87|issue=1|year=2000|pages=145–156|issn=0006-3444|doi=10.1093/biomet/87.1.145}}</ref><ref name="Cramer1976">{{cite journal|last1=Cramer|first1=Harald|title=Half a Century with Probability Theory: Some Personal Recollections|journal=The Annals of Probability|volume=4|issue=4|year=1976|pages=509–546|issn=0091-1798|doi=10.1214/aop/1176996025|doi-access=free}}</ref> Decades later Cramér referred to the 1930s as the "heroic period of mathematical probability theory".<ref name="Cramer1976"/>

In mathematics, the theory of stochastic processes is an important contribution to [[probability theory]],<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 continues to be an active topic of research for both theory 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&pg=PR5|year=2011|publisher=European Mathematical Society|isbn=978-3-03719-072-2|pages=5–}}</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|date=12 February 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|date=22 August 2014|publisher=Springer|isbn=978-3-319-08488-6|pages=vii–ix}}</ref>

The word ''stochastic'' is used to describe other terms and objects in mathematics. Examples include a [[stochastic matrix]], which describes a stochastic process known as a [[Markov process]], and stochastic calculus, which involves [[differential equation]]s and  [[integral]]s based on stochastic processes such as the [[Wiener process]], also called the Brownian motion process.