Editing Martingale (probability theory) (section)

==History==
Originally, ''[[martingale (betting system)|martingale]]'' referred to a class of [[betting strategy|betting strategies]] that was popular in 18th-century [[France]].<ref>{{cite book| first=N. J. |last=Balsara|title=Money Management Strategies for Futures Traders|publisher= Wiley Finance|year= 1992| isbn =978-0-471-52215-7 |page=[https://archive.org/details/moneymanagements00bals/page/122 122]|url=https://archive.org/details/moneymanagements00bals| url-access=registration | quote=martingale. }}</ref><ref>{{cite journal|url=http://www.jehps.net/juin2009/Mansuy.pdf|title=The origins of the Word "Martingale"|last1=Mansuy|first1=Roger|date=June 2009|volume=5|number=1|journal=Electronic Journal for History of Probability and Statistics|access-date=2011-10-22|archive-url=https://web.archive.org/web/20120131103618/http://www.jehps.net/juin2009/Mansuy.pdf|archive-date=2012-01-31|url-status=live}}</ref>  The simplest of these strategies was designed for a game in which the [[gambler]] wins their stake if a coin comes up heads and loses it if the coin comes up tails.  The strategy had the gambler double their bet after every loss so that the first win would recover all previous losses plus win a profit equal to the original stake.  As the gambler's wealth and available time jointly approach infinity, their probability of eventually flipping heads approaches 1, which makes the martingale betting strategy seem like a [[almost surely|sure thing]].  However, the [[exponential growth]] of the bets eventually bankrupts its users due to finite bankrolls. [[Stopped process#Brownian motion|Stopped Brownian motion]], which is a martingale process, can be used to model the trajectory of such games.

The concept of martingale in probability theory was introduced by [[Paul Lévy (mathematician)|Paul Lévy]] in 1934, though he did not name it. The term "martingale" was introduced later by {{harvtxt|Ville|1939}}, who also extended the definition to continuous martingales. Much of the original development of the theory was done by [[Joseph Leo Doob]] among others.  Part of the motivation for that work was to show the impossibility of successful betting strategies in games of chance.