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Concept
Continuous stochastic process
Summary
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 another terminology, this would be a continuous-time stochastic process, in parallel to a "discrete-time process". Given the possible confusion, caution is needed.
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.
Given a time t ∈ T, X is said to be continuous with probability one at t if
Given a time t ∈ T, X is said to be continuous in mean-square at t if E[|Xt|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
Given a time t ∈ T, X is said to be continuous in distribution at t if
for all points x at which Ft is continuous, where Ft denotes the cumulative distribution function of the random variable Xt.
Sample continuous process
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-continuous process
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.
Official source
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