Continuous stochastic process

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 mutandisifSisRⁿ, a normed vector space, or even a general metric space.

Given a time t ∈ T, X is said to be continuous with probability oneattif

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

Given a time t ∈ T, X is said to be continuous in mean-squareattifE[|X_t|²] < +∞ and

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

Given a time t ∈ T, X is said to be continuous in probabilityatt 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 tif

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

Given a time t ∈ T, X is said to be continuous in distributionattif

${\displaystyle \lim _{s\to t}F_{s}($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.

X is said to be sample continuousifX_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.

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.

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.

• 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)