Continuous stochastic process

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.

Continuity almost surely

Given a time t ∈ T, X is said to be continuous with probability one at t if

${\displaystyle \mathbf{P}\left(\left\{\omega \in \mathrm{\Omega }|\underset{s\to t}{lim}|X_s(\omega )-X_t(\omega )|=0\right\}\right)=1.}$

Mean-square continuity

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

${\displaystyle \underset{s\to t}{lim}\mathbf{E}\left[|X_s-X_t{|}^2\right]=0.}$

Continuity in probability

Given a time t ∈ T, X is said to be continuous in probability at t if, for all ε > 0,

${\displaystyle \underset{s\to t}{lim}\mathbf{P}\left(\left\{\omega \in \mathrm{\Omega }||X_s(\omega )-X_t(\omega )|\ge \epsilon \right\}\right)=0.}$

or equivalently

${\displaystyle \underset{s\to t}{lim}\mathbf{P}\left(\left\{\omega \in \mathrm{\Omega }||X_s(\omega )-X_t(\omega )|<\epsilon \right\}\right)=1.}$

Also equivalent, X is continuous in probability at time t if

${\displaystyle \underset{s\to t}{lim}\mathbf{E}\left[\frac{|X_s-X_t|}{1+|X_s-X_t|}\right]=0.}$

Continuity in distribution

Given a time t ∈ T, X is said to be continuous in distribution at t if

${\displaystyle \underset{s\to t}{lim}{F}_s(x)=F_t(x)}$

for all points x at which F_t is continuous, where F_t denotes the cumulative distribution function of the random variable X_t.

Sample continuity

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

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.