Continuous stochastic process

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In the theory of stochastic processes, there are many senses in which a process may be said to be "continuous" as a function of its "time" 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 analyse.

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[edit] 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.

[edit] 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.

[edit] 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.

[edit] 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.

[edit] 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.

[edit] Sample continuity

Main article: 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.

[edit] Feller continuity

Main article: 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. Here x denotes the initial state of the process X, and Ex denotes expectation conditional upon the event that X starts at x.

[edit] 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}.

Note that 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.

[edit] References

  • 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 edition, Berlin: Springer. ISBN 3-540-04758-1.  (See Lemma 8.1.4)