Editing Stochastic process (section)

===Lévy process===
{{Main|Lévy process}}
Lévy processes are types of stochastic processes that can be considered as generalizations of random walks in continuous time.<ref name="Applebaum2004page1337"/><ref name="Bertoin1998pageVIII">{{cite book|author=Jean Bertoin|title=Lévy Processes|url=https://books.google.com/books?id=ftcsQgMp5cUC&pg=PR8|year=1998|publisher=Cambridge University Press|isbn=978-0-521-64632-1|page=viii}}</ref> These processes have many applications in fields such as finance, fluid mechanics, physics and biology.<ref name="Applebaum2004page1336">{{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}}</ref><ref name="ApplebaumBook2004page69">{{cite book|author=David Applebaum|title=Lévy Processes and Stochastic Calculus|url=https://books.google.com/books?id=q7eDUjdJxIkC|year=2004|publisher=Cambridge University Press|isbn=978-0-521-83263-2|page=69}}</ref> The main defining characteristics of these processes are their stationarity and independence properties, so they were known as ''processes with stationary and independent increments''. In other words, a stochastic process <math>X</math> is a Lévy process if for <math>n</math> non-negatives numbers, <math>0\leq t_1\leq \dots \leq t_n</math>, the corresponding <math>n-1</math> increments
<div class="center"><math>
X_{t_2}-X_{t_1}, \dots ,  X_{t_n}-X_{t_{n-1}},
</math></div>
are all independent of each other, and the distribution of each increment only depends on the difference in time.<ref name="Applebaum2004page1337"/>

A Lévy process can be defined such that its state space is some abstract mathematical space, such as a [[Banach space]], but the processes are often defined so that they take values in Euclidean space. The index set is the non-negative numbers, so <math> I= [0,\infty) </math>, which gives the interpretation of time. Important stochastic processes such as the Wiener process, the homogeneous Poisson process (in one dimension), and [[subordinator (mathematics)|subordinators]] are all Lévy processes.<ref name="Applebaum2004page1337"/><ref name="Bertoin1998pageVIII"/>