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 wellbehaved in some sense, and, therefore, much easier to analyse. It is implicit here that the index of the stochastic process is a continuous variable. Note that some authors^{[1]} define a "continuous (stochastic) process" as only requiring that the index variable be continuous, without continuity of sample paths: in some terminology, this would be a continuoustime stochastic process, in parallel to a "discretetime process". Given the possible confusion, caution is needed.^{[1]}
Contents

Definitions 1

Continuity with probability one 1.1

Meansquare continuity 1.2

Continuity in probability 1.3

Continuity in distribution 1.4

Sample continuity 1.5

Feller continuity 1.6

Relationships 2

Notes 3

References 4
Definitions
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 R^{n}, a normed vector space, or even a general metric space.
Continuity with probability one
Given a time t ∈ T, X is said to be continuous with probability one at t if

\mathbf{P} \left( \left\{ \omega \in \Omega \left \lim_{s \to t} \big X_{s} (\omega)  X_{t} (\omega) \big = 0 \right. \right\} \right) = 1.
Meansquare continuity
Given a time t ∈ T, X is said to be continuous in meansquare at t if E[X_{t}^{2}] < +∞ and

\lim_{s \to t} \mathbf{E} \left[ \big X_{s}  X_{t} \big^{2} \right] = 0.
Continuity in probability
Given a time t ∈ T, X is said to be continuous in probability at t if, for all ε > 0,

\lim_{s \to t} \mathbf{P} \left( \left\{ \omega \in \Omega \left \big X_{s} (\omega)  X_{t} (\omega) \big \geq \varepsilon \right. \right\} \right) = 0.
Equivalently, X is continuous in probability at time t if

\lim_{s \to t} \mathbf{E} \left[ \frac{\big X_{s}  X_{t} \big}{1 + \big X_{s}  X_{t} \big} \right] = 0.
Continuity in distribution
Given a time t ∈ T, X is said to be continuous in distribution at t if

\lim_{s \to t} F_{s} (x) = F_{t} (x)
for all points x at which F_{t} is continuous, where F_{t} denotes the cumulative distribution function of the random variable X_{t}.
Sample continuity
X is said to be sample continuous if X_{t}(ω) is continuous in t for Palmost all ω ∈ Ω. Sample continuity is the appropriate notion of continuity for processes such as Itō diffusions.
Feller continuity
X is said to be a Fellercontinuous process if, for any fixed t ∈ T and any bounded, continuous and Σmeasurable function g : S → R, E^{x}[g(X_{t})] depends continuously upon x. Here x denotes the initial state of the process X, and E^{x} denotes expectation conditional upon the event that X starts at x.
Relationships
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:

continuity with probability one implies continuity in probability;

continuity in meansquare implies continuity in probability;

continuity with probability one neither implies, nor is implied by, continuity in meansquare;

continuity in probability implies, but is not implied by, continuity in distribution.
It is tempting to confuse continuity with probability one with sample continuity. Continuity with probability one at time t means that P(A_{t}) = 0, where the event A_{t} is given by

A_{t} = \left\{ \omega \in \Omega \left \lim_{s \to t} \big X_{s} (\omega)  X_{t} (\omega) \big \neq 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 = \bigcup_{t \in T} A_{t}.
Note that 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(A_{t}) = 0 for every t ∈ T. This is the case, for example, with the telegraph process.
Notes

^ ^{a} ^{b} Dodge, Y. (2006) The Oxford Dictionary of Statistical Terms, OUP. ISBN 0199206139 (Entry for "continuous process")
References

Kloeden, Peter E.; Platen, Eckhard (1992). Numerical solution of stochastic differential equations. Applications of Mathematics (New York) 23. Berlin: SpringerVerlag. pp. 38–39;.

(See Lemma 8.1.4)
This article was sourced from Creative Commons AttributionShareAlike License; additional terms may apply. World Heritage Encyclopedia content is assembled from numerous content providers, Open Access Publishing, and in compliance with The Fair Access to Science and Technology Research Act (FASTR), Wikimedia Foundation, Inc., Public Library of Science, The Encyclopedia of Life, Open Book Publishers (OBP), PubMed, U.S. National Library of Medicine, National Center for Biotechnology Information, U.S. National Library of Medicine, National Institutes of Health (NIH), U.S. Department of Health & Human Services, and USA.gov, which sources content from all federal, state, local, tribal, and territorial government publication portals (.gov, .mil, .edu). Funding for USA.gov and content contributors is made possible from the U.S. Congress, EGovernment Act of 2002.
Crowd sourced content that is contributed to World Heritage Encyclopedia is peer reviewed and edited by our editorial staff to ensure quality scholarly research articles.
By using this site, you agree to the Terms of Use and Privacy Policy. World Heritage Encyclopedia™ is a registered trademark of the World Public Library Association, a nonprofit organization.