Equilibrium unfolding

2013-11-19T07:56:52Z

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In [[mathematics]], a '''sample-continuous process''' is a [[stochastic process]] whose sample paths are [[almost surely]] [[continuous function]]s.

==Definition==

Let (Ω, Σ, '''P''') be a [[probability space]]. Let ''X'' : ''I'' × Ω → ''S'' be a stochastic process, where the [[index set]] ''I'' and state space ''S'' are both [[topological space]]s. Then the process ''X'' is called '''sample-continuous''' (or '''almost surely continuous''', or simply '''continuous''') if the map ''X''(''ω'') : ''I'' → ''S'' is [[Continuous function (topology)|continuous as a function of topological spaces]] for '''P'''-[[almost all]] ''ω'' in ''Ω''.

In many examples, the index set ''I'' is an interval of time, [0, ''T''] or [0, +∞), and the state space ''S'' is the [[real line]] or ''n''-[[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]]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'' : [0, +∞) × Ω → '''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]]s determine the [[Law (stochastic processes)|law]], and vice versa.

==See also==

* [[Continuous stochastic process]]

==References==

* {{cite book
| author = Kloeden, Peter E.
|coauthors = 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
}}

[[Category:Stochastic processes]]

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Equilibrium unfolding