Sample From a Continuous Process Definition

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 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]

Definitions [edit]

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 [edit]

Given a time t ∈T, X is said to be continuous with probability one at t if

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

Mean-square continuity [edit]

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

${\displaystyle \lim _{s\to t}\mathbf {E} \left[{\big |}X_{s}-X_{t}{\big |}^{2}\right]=0.}$

Continuity in probability [edit]

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

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

Equivalently, X is continuous in probability at time t if

${\displaystyle \lim _{s\to t}\mathbf {E} \left[{\frac {{\big |}X_{s}-X_{t}{\big |}}{1+{\big |}X_{s}-X_{t}{\big |}}}\right]=0.}$

Continuity in distribution [edit]

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

${\displaystyle \lim _{s\to t}F_{s}(x)=F_{t}(x)}$

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 [edit]

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 [edit]

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 [edit]

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

${\displaystyle A_{t}=\left\{\omega \in \Omega \left|\lim _{s\to t}{\big |}X_{s}(\omega )-X_{t}(\omega ){\big |}\neq 0\right.\right\},}$

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

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

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(A _t ) = 0 for every t ∈T. This is the case, for example, with the telegraph process.

Notes [edit]

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

References [edit]

• Kloeden, Peter E.; Platen, Eckhard (1992). Numerical solution of stochastic differential equations. Applications of Mathematics (New York) 23. Berlin: Springer-Verlag. pp. 38–39. ISBN3-540-54062-8.
• Øksendal, Bernt K. (2003). Stochastic Differential Equations: An Introduction with Applications (Sixth ed.). Berlin: Springer. ISBN3-540-04758-1. (See Lemma 8.1.4)

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Source: https://en.wikipedia.org/wiki/Continuous_stochastic_process

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