# 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 analyze. It is implicit here that the index of the stochastic process is a continuous variable. Some authors 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.

## 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 Rn, 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

$\mathbf {P} \left(\left\{\omega \in \Omega \left|\lim _{s\to t}{\big |}X_{s}(\omega )-X_{t}(\omega ){\big |}=0\right.\right\}\right)=1.$

### Mean-square continuity

Given a time t ∈ T, X is said to be continuous in mean-square at t if E[|Xt|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 Ft is continuous, where Ft denotes the cumulative distribution function of the random variable Xt.

### Sample continuity

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.

### Feller continuity

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(Xt)] 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.

## 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(At) = 0, where the event At 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

$A=\bigcup _{t\in T}A_{t}.$

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.