Continuous stochastic process (original) (raw)

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Stochastic process that is a continuous function of time or index parameter

In probability theory, a continuous stochastic process is a type of stochastic process that may be said to be "continuous" as a function of its "time" or index 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 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]

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

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

P ( { ω ∈ Ω | lim s → t | X s ( ω ) − X t ( ω ) | = 0 } ) = 1. {\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.} {\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.}

Mean-square continuity

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

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

Continuity in probability

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

lim s → t P ( { ω ∈ Ω | | X s ( ω ) − X t ( ω ) | ≥ 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.} {\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 t if

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

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

lim s → t F s ( x ) = F t ( x ) {\displaystyle \lim _{s\to t}F_{s}(x)=F_{t}(x)} {\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 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.

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 Ex 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:

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 = { ω ∈ Ω | lim s → t | X s ( ω ) − X t ( ω ) | ≠ 0 } , {\displaystyle A_{t}=\left\{\omega \in \Omega \left|\lim _{s\to t}{\big |}X_{s}(\omega )-X_{t}(\omega ){\big |}\neq 0\right.\right\},} {\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

A = ⋃ t ∈ T A t . {\displaystyle A=\bigcup _{t\in T}A_{t}.} {\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.

  1. ^ a b Dodge, Y. (2006) The Oxford Dictionary of Statistical Terms, OUP. ISBN 0-19-920613-9 (Entry for "continuous process")