# Continuous 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 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 **continuous-time stochastic process**, in parallel to a "discrete-time process". Given the possible confusion, caution is needed.^{[1]}

## Contents

## 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

### Mean-square continuity

Given a time *t* ∈ *T*, *X* is said to be **continuous in mean-square** at *t* if **E**[|*X*_{t}|^{2}] < +∞ and

### Continuity in probability

Given a time *t* ∈ *T*, *X* is said to be **continuous in probability** at *t* if, for all *ε* > 0,

Equivalently, *X* is continuous in probability at time *t* if

### Continuity in distribution

Given a time *t* ∈ *T*, *X* is said to be **continuous in distribution** at *t* if

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

{{#invoke:main|main}}

*X* is said to be **sample continuous** if *X*_{t}(*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.

### Feller continuity

{{#invoke:main|main}}

*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**, **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 mean-square implies continuity in probability;
- continuity with probability one neither implies, nor is implied by, continuity in mean-square;
- 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

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

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

- ↑
^{1.0}^{1.1}Dodge, Y. (2006)*The Oxford Dictionary of Statistical Terms*, OUP. ISBN 0-19-920613-9 (Entry for "continuous process")

## References

- {{#invoke:citation/CS1|citation

|CitationClass=book }}

- {{#invoke:citation/CS1|citation

|CitationClass=book }} (See Lemma 8.1.4)