Continuous stochastic process

In

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^[1] define a "continuous (stochastic) process" as only requiring that the index variable be continuous, without continuity of sample paths: in another terminology, this would be a continuous-time stochastic process, in parallel to a "discrete-time process". Given the possible confusion, caution is needed.^[1]

Definitions

Let (Ω, Σ, P) be 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

\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.

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

\lim _{s\to t}\mathbf {E} \left[{\big |}X_{s}-X_{t}{\big |}^{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,

\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.

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 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-

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.

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(A_t) = 0, where the event A_t 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}.

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

.

Notes

^
ISBN 0-19-920613-9
(Entry for "continuous process")

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
.

ISBN 3-540-04758-1
. (See Lemma 8.1.4)

[D-1] 
ISBN 0-19-920613-9
(Entry for "continuous process")

[1]

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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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