Sample-continuous process

In mathematics, a sample-continuous process is a stochastic process whose sample paths are almost surely continuous functions.

Definition

Let (Ω, Σ, P) be a

continuous as a function of topological spaces for P-almost all

ω in Ω.

In many examples, the index set I is an interval of time, [0, T] or [0, +∞), and the state space S is the

real line or n-dimensional Euclidean space

Rⁿ.

Examples

Brownian motion (the Wiener process) on Euclidean space is sample-continuous.
For "nice" parameters of the equations, solutions to stochastic differential equations are sample-continuous. See the existence and uniqueness theorem in the stochastic differential equations article for some sufficient conditions to ensure sample continuity.
The process X : [0, +∞) × Ω → R that makes equiprobable jumps up or down every unit time according to

{\begin{cases}X_{t}\sim \mathrm {Unif} (\{X_{t-1}-1,X_{t-1}+1\}),&t{\mbox{ an integer;}}\\X_{t}=X_{\lfloor t\rfloor },&t{\mbox{ not an integer;}}\end{cases}}

is not sample-continuous. In fact, it is surely discontinuous.

Properties

For sample-continuous processes, the finite-dimensional distributions determine the law, and vice versa.

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
.

Discrete time		Bernoulli process Branching process Chinese restaurant process Galton–Watson process Independent and identically distributed random variables Markov chain Moran process Random walk Loop-erased Self-avoiding Biased Maximal entropy
Continuous time	Additive process Bessel process Birth–death process pure birth Brownian motion Bridge Excursion Fractional Geometric Meander Cauchy process Contact process Continuous-time random walk Cox process Diffusion process Dyson Brownian motion Empirical process Feller process Fleming–Viot process Gamma process Geometric process Hawkes process Hunt process Interacting particle systems Itô diffusion Itô process Jump diffusion Jump process Lévy process Local time Markov additive process McKean–Vlasov process Ornstein–Uhlenbeck process Poisson process Compound Non-homogeneous Schramm–Loewner evolution Semimartingale Sigma-martingale Stable process Superprocess Telegraph process Variance gamma process Wiener process Wiener sausage
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List of topics Category

Definition

Examples

Properties

See also

References