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.
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
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
Continuity in distribution
Given a time t ∈ T, X is said to be continuous in distribution at t if
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.
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
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(At) = 0 for every t ∈ T. This is the case, for example, with the telegraph process.
This article includes a list of references, but its sources remain unclear because it has insufficient inline citations. (November 2010) (Learn how and when to remove this template message)
- Kloeden, Peter E.; Platen, Eckhard (1992). Numerical solution of stochastic differential equations. Applications of Mathematics (New York) 23. Berlin: Springer-Verlag. pp. 38–39;. ISBN 3-540-54062-8.
- Øksendal, Bernt K. (2003). Stochastic Differential Equations: An Introduction with Applications (Sixth ed.). Berlin: Springer. ISBN 3-540-04758-1. (See Lemma 8.1.4)