View source for Sample-continuous process

In [[Mathematics|mathematics]], a '''sample-continuous process''' is a [[Stochastic process|stochastic process]] whose sample paths are [[Almost surely|almost surely]] [[Continuous function|continuous function]]s.
==Definition==

Let (&Omega;,&nbsp;&Sigma;,&nbsp;'''P''') be a [[Probability space|probability space]]. Let ''X''&nbsp;:&nbsp;''I''&nbsp;&times;&nbsp;&Omega;&nbsp;&rarr;&nbsp;''S'' be a stochastic process, where the [[Index set|index set]] ''I'' and state space ''S'' are both [[Topological space|topological space]]s. Then the process ''X'' is called '''sample-continuous''' (or '''almost surely continuous''', or simply '''continuous''') if the map ''X''(''&omega;'')&nbsp;:&nbsp;''I''&nbsp;&rarr;&nbsp;''S'' is continuous as a function of topological spaces for '''P'''-[[Almost all|almost all]] ''&omega;'' in ''&Omega;''.

In many examples, the index set ''I'' is an interval of time, [0,&nbsp;''T''] or [0,&nbsp;+&infin;), and the state space ''S'' is the [[Real line|real line]] or ''n''-[[Physics:Dimension|dimension]]al [[Euclidean space]] '''R'''<sup>''n''</sup>.

==Examples==

* [[Brownian motion]] (the [[Wiener process]]) on Euclidean space is sample-continuous.
* For "nice" parameters of the equations, solutions to [[Stochastic differential equation|stochastic differential equation]]s are sample-continuous. See the existence and uniqueness theorem in the stochastic differential equations article for some sufficient conditions to ensure sample continuity.
* The process ''X''&nbsp;:&nbsp;[0,&nbsp;+&infin;)&nbsp;&times;&nbsp;&Omega;&nbsp;&rarr;&nbsp;'''R''' that makes equiprobable jumps up or down every unit time according to

::<math>\begin{cases} X_{t} \sim \mathrm{Unif} (\{X_{t-1} - 1, X_{t-1} + 1\}), & t \mbox{ an integer;} \\ X_{t} = X_{\lfloor t \rfloor}, & t \mbox{ not an integer;} \end{cases}</math>

: is ''not'' sample-continuous. In fact, it is surely discontinuous.

==Properties==

* For sample-continuous processes, the [[Finite-dimensional distribution|finite-dimensional distribution]]s determine the [[Law (stochastic processes)|law]], and vice versa.

==See also==

* [[Continuous stochastic process]]

==References==

* {{cite book
|   author = Kloeden, Peter E.
|author2=Platen, Eckhard
 |    title = Numerical solution of stochastic differential equations
|   series = Applications of Mathematics (New York) 23
|publisher = Springer-Verlag
| location = Berlin
|     year = 1992
|    pages = 38–39
|     isbn = 3-540-54062-8
}}

{{Stochastic processes}}

[[Category:Stochastic processes]]

{{Sourceattribution|Sample-continuous process}}