Editing Stochastic process (section)

===Martingale===
{{Main|Martingale (probability theory)}}
A martingale is a discrete-time or continuous-time stochastic process with the property that, at every instant, given the current value and all the past values of the process, the conditional expectation of every future value is equal to the current value. In discrete time, if this property holds for the next value, then it holds for all future values. The exact mathematical definition of a martingale requires two other conditions coupled with the mathematical concept of a filtration, which is related to the intuition of increasing available information as time passes. Martingales are usually defined to be real-valued,<ref name="Klebaner2005page65">{{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=65}}</ref><ref name="KaratzasShreve2014page11">{{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=11}}</ref><ref name="Williams1991page93">{{cite book|author=David Williams|title=Probability with Martingales|url=https://books.google.com/books?id=e9saZ0YSi-AC|year=1991|publisher=Cambridge University Press|isbn=978-0-521-40605-5|pages=93, 94}}</ref> but they can also be complex-valued<ref name="Doob1990page292">{{cite book|author=Joseph L. Doob|title=Stochastic processes|url=https://books.google.com/books?id=NrsrAAAAYAAJ|year=1990|publisher=Wiley|pages=292, 293}}</ref> or even more general.<ref name="Pisier2016">{{cite book|author=Gilles Pisier|title=Martingales in Banach Spaces|url=https://books.google.com/books?id=n3JNDAAAQBAJ&pg=PR4|year=2016|publisher=Cambridge University Press|isbn=978-1-316-67946-3}}</ref>

A symmetric random walk and a Wiener process (with zero drift) are both examples of martingales, respectively, in discrete and continuous time.<ref name="Klebaner2005page65"/><ref name="KaratzasShreve2014page11"/> For a [[sequence]] of [[independent and identically distributed]] random variables <math>X_1, X_2, X_3, \dots</math> with zero mean, the stochastic process formed from the successive partial sums <math>X_1,X_1+ X_2, X_1+ X_2+X_3, \dots</math> is a discrete-time martingale.<ref name="Steele2012page12">{{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|pages=12, 13}}</ref> In this aspect, discrete-time martingales generalize the idea of partial sums of independent random variables.<ref name="HallHeyde2014page2">{{cite book|author1=P. Hall|author2-link=Chris Heyde|author2=C. C. Heyde|title=Martingale Limit Theory and Its Application|url=https://books.google.com/books?id=gqriBQAAQBAJ&pg=PR10|year=2014|publisher=Elsevier Science|isbn=978-1-4832-6322-9|page=2}}</ref>

Martingales can also be created from stochastic processes by applying some suitable transformations, which is the case for the homogeneous Poisson process (on the real line) resulting in a martingale called the ''compensated Poisson process''.<ref name="KaratzasShreve2014page11"/> Martingales can also be built from other martingales.<ref name="Steele2012page12"/> For example, there are martingales based on the martingale the Wiener process, forming continuous-time martingales.<ref name="Klebaner2005page65"/><ref name="Steele2012page115">{{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=115}}</ref>

Martingales mathematically formalize the idea of a 'fair game' where it is possible form reasonable expectations for payoffs,<ref name="Ross1996page295">{{cite book|author=Sheldon M. Ross|title=Stochastic processes|url=https://books.google.com/books?id=ImUPAQAAMAAJ|year=1996|publisher=Wiley|isbn=978-0-471-12062-9|page=295}}</ref> and they were originally developed to show that it is not possible to gain an 'unfair' advantage in such a game.<ref name="Steele2012page11"/> But now they are used in many areas of probability, which is one of the main reasons for studying them.<ref name="Williams1991page93"/><ref name="Steele2012page11">{{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=11}}</ref><ref name="Kallenberg2002page96">{{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=96}}</ref> Many problems in probability have been solved by finding a martingale in the problem and studying it.<ref name="Steele2012page371">{{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=371}}</ref> Martingales will converge, given some conditions on their moments, so they are often used to derive convergence results, due largely to [[martingale convergence theorem]]s.<ref name="HallHeyde2014page2"/><ref name="Steele2012page22">{{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=22}}</ref><ref name="GrimmettStirzaker2001page336">{{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=336}}</ref>

Martingales have many applications in statistics, but it has been remarked that its use and application are not as widespread as it could be in the field of statistics, particularly statistical inference.<ref name="GlassermanKou2006">{{cite journal|last1=Glasserman|first1=Paul|last2=Kou|first2=Steven|title=A Conversation with Chris Heyde|journal=Statistical Science|volume=21|issue=2|year=2006|pages=292, 293|issn=0883-4237|doi=10.1214/088342306000000088|arxiv=math/0609294|bibcode=2006math......9294G|s2cid=62552177}}</ref> They have found applications in areas in probability theory such as queueing theory and Palm calculus<ref name="BaccelliBremaud2013">{{cite book|author1=Francois Baccelli|author2=Pierre Bremaud|title=Elements of Queueing Theory: Palm Martingale Calculus and Stochastic Recurrences|url=https://books.google.com/books?id=DH3pCAAAQBAJ&pg=PR2|year=2013|publisher=Springer Science & Business Media|isbn=978-3-662-11657-9}}</ref> and other fields such as economics<ref name="HallHeyde2014pageX">{{cite book|author1=P. Hall|author2=C. C. Heyde|title=Martingale Limit Theory and Its Application|url=https://books.google.com/books?id=gqriBQAAQBAJ&pg=PR10|year= 2014|publisher=Elsevier Science|isbn=978-1-4832-6322-9|page=x}}</ref> and finance.<ref name="MusielaRutkowski2006"/>