View source for Continuous-time stochastic process

In [[probability theory]] and [[statistics]], a '''continuous-time stochastic process''', or a '''continuous-space-time stochastic process''' is a [[stochastic process]] for which the index variable takes a continuous set of values, as contrasted with a [[discrete-time signal|discrete-time process]] for which the index variable takes only distinct values. An alternative terminology uses '''continuous parameter''' as being more inclusive.<ref>Parzen, E. (1962) ''Stochastic Processes'', Holden-Day. {{ISBN|0-8162-6664-6}} (Chapter 6)</ref>

A more restricted class of processes are the [[continuous stochastic process]]es; here the term often (but not always<ref name=D>Dodge, Y. (2006) ''The Oxford Dictionary of Statistical Terms'', OUP. {{ISBN|0-19-920613-9}} (Entry for "continuous process")</ref>) implies both that the index variable is continuous and that sample paths of the process are continuous. Given the possible confusion, caution is needed.<ref name=D/>

Continuous-time stochastic processes that are constructed from discrete-time processes via a waiting time distribution are called [[continuous-time random walk]]s.<ref>{{cite book|last1=Paul|first1=Wolfgang|last2=Baschnagel|first2=Jörg|title=Stochastic Processes: From Physics to Finance|url=https://books.google.com/books?id=OWANAAAAQBAJ&pg=PA72|accessdate=20 June 2022|date=2013-07-11|publisher=Springer Science & Business Media|isbn=9783319003276|pages=72–74}}</ref>

==Examples==

An example of a continuous-time stochastic process for which sample paths are not continuous is a [[Poisson process]]. An example with continuous paths is the [[Ornstein–Uhlenbeck process]].

==See also==
* [[Continuous signal]]

==References==
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{{Stochastic processes|state=collapsed}}

[[Category:Stochastic processes]]