Editing Stochastic process (section)

===Random walk===
{{Main|Random walk}}

[[Random walks]] are stochastic processes that are usually defined as sums of [[iid]] random variables or random vectors in Euclidean space, so they are processes that change in discrete time.<ref name="Klenke2013page347">{{cite book|author=Achim Klenke|title=Probability Theory: A Comprehensive Course|url=https://books.google.com/books?id=aqURswEACAAJ|year=2013|publisher=Springer|isbn=978-1-4471-5362-7|pages=347}}</ref><ref name="LawlerLimic2010page1">{{cite book|author1=Gregory F. Lawler|author2=Vlada Limic|title=Random Walk: A Modern Introduction|url=https://books.google.com/books?id=UBQdwAZDeOEC|year=2010|publisher=Cambridge University Press|isbn=978-1-139-48876-1|page=1}}</ref><ref name="Kallenberg2002page136">{{cite book|author=Olav Kallenberg|title=Foundations of Modern Probability|url=https://books.google.com/books?id=L6fhXh13OyMC|date= 2002|publisher=Springer Science & Business Media|isbn=978-0-387-95313-7|page=136}}</ref><ref name="Florescu2014page383">{{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|page=383}}</ref><ref name="Durrett2010page277">{{cite book|author=Rick Durrett|title=Probability: Theory and Examples|url=https://books.google.com/books?id=evbGTPhuvSoC|year=2010|publisher=Cambridge University Press|isbn=978-1-139-49113-6|page=277}}</ref> But some also use the term to refer to processes that change in continuous time,<ref name="Weiss2006page1">{{cite book|last1=Weiss|first1=George H.|title=Encyclopedia of Statistical Sciences|chapter=Random Walks|year=2006|doi=10.1002/0471667196.ess2180.pub2|page=1|isbn=978-0471667193}}</ref> particularly the Wiener process used in financial models, which has led to some confusion, resulting in its criticism.<ref name="Spanos1999page454">{{cite book|author=Aris Spanos|title=Probability Theory and Statistical Inference: Econometric Modeling with Observational Data|url=https://books.google.com/books?id=G0_HxBubGAwC|year=1999|publisher=Cambridge University Press|isbn=978-0-521-42408-0|page=454}}</ref> There are various other types of random walks, defined so their state spaces can be other mathematical objects, such as lattices and groups, and in general they are highly studied and have many applications in different disciplines.<ref name="Weiss2006page1"/><ref name="Klebaner2005page81">{{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=81}}</ref>

A classic example of a random walk is known as the ''simple random walk'', which is a stochastic process in discrete time with the integers as the state space, and is based on a Bernoulli process, where each Bernoulli variable takes either the value positive one or negative one. In other words, the simple random walk takes place on the integers, and its value increases by one with probability, say, <math>p</math>, or decreases by one with probability <math>1-p</math>, so the index set of this random walk is the natural numbers, while its state space is the integers. If <math>p=0.5</math>, this random walk is called a symmetric random walk.<ref name="Gut2012page88">{{cite book|author=Allan Gut|title=Probability: A Graduate Course|url=https://books.google.com/books?id=XDFA-n_M5hMC|year=2012|publisher=Springer Science & Business Media|isbn=978-1-4614-4708-5|page=88}}</ref><ref name="GrimmettStirzaker2001page71">{{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=71}}</ref>