When is a Random Process Continuous Continuous Valued Random 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 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
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
Continuity in probability [edit]
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 [edit]
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 [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
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 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]
- ^ 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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