Continuous stochastic process should not be confused with Continuous-time stochastic process.
In probability theory, a continuous stochastic process is a type of stochastic process that may be said to be "continuous" as a function of its "time" or index parameter. Continuity is a nice property for (the sample paths of) a process to have, since it implies that they are well-behaved in some sense, and, therefore, much easier to analyze. It is implicit here that the index of the stochastic process is a continuous variable. Some authors[1] define a "continuous (stochastic) process" as only requiring that the index variable be continuous, without continuity of sample paths: in another terminology, this would be a continuous-time stochastic process, in parallel to a "discrete-time process". Given the possible confusion, caution is needed.[1]
Let (Ω, Σ, P) be a probability space, let T be some interval of time, and let X : T × Ω → S be a stochastic process. For simplicity, the rest of this article will take the state space S to be the real line R, but the definitions go through mutatis mutandis if S is Rn, a normed vector space, or even a general metric space.
Given a time t ∈ T, X is said to be continuous with probability one at t if
P\left(\left\{\omega\in\Omega\left|\lims|Xs(\omega)-Xt(\omega)|=0\right.\right\}\right)=1.
Given a time t ∈ T, X is said to be continuous in mean-square at t if E[|''X''<sub>''t''</sub>|<sup>2</sup>] < +∞ and
\limsE\left[|Xs-Xt|2\right]=0.
See main article: article and Continuity in probability. Given a time t ∈ T, X is said to be continuous in probability at t if, for all ε > 0,
\limsP\left(\left\{\omega\in\Omega\left||Xs(\omega)-Xt(\omega)|\geq\varepsilon\right.\right\}\right)=0.
Equivalently, X is continuous in probability at time t if
\limsE\left[
| |Xs-Xt| | |
| 1+|Xs-Xt| |
\right]=0.
Given a time t ∈ T, X is said to be continuous in distribution at t if
\limsFs(x)=Ft(x)
for all points x at which Ft is continuous, where Ft denotes the cumulative distribution function of the random variable Xt.
See main article: Sample continuous process.
X is said to be sample continuous if Xt(ω) is continuous in t for P-almost all ω ∈ Ω. Sample continuity is the appropriate notion of continuity for processes such as Itō diffusions.
See main article: Feller-continuous process.
X is said to be a Feller-continuous process if, for any fixed t ∈ T and any bounded, continuous and Σ-measurable function g : S → R, Ex[''g''(''X''<sub>''t''</sub>)] depends continuously upon x. Here x denotes the initial state of the process X, and Ex denotes expectation conditional upon the event that X starts at x.
The relationships between the various types of continuity of stochastic processes are akin to the relationships between the various types of convergence of random variables. In particular:
It is tempting to confuse continuity with probability one with sample continuity. Continuity with probability one at time t means that P(At) = 0, where the event At is given by
At=\left\{\omega\in\Omega\left|\lims|Xs(\omega)-Xt(\omega)| ≠ 0\right.\right\},
and it is perfectly feasible to check whether or not this holds for each t ∈ T. Sample continuity, on the other hand, requires that P(A) = 0, where
A=cuptAt.
A is an uncountable union of events, so it may not actually be an event itself, so P(A) may be undefined! Even worse, even if A is an event, P(A) can be strictly positive even if P(At) = 0 for every t ∈ T. This is the case, for example, with the telegraph process.