Continuous stochastic process

<h2 id="definitions">Definitions</h2>
Let (Ω, Σ, P) be a <a href="/facts/Probability_space/dEcv2VrU">probability space</a>, let T be some <a href="/facts/Interval_(mathematics)/e9se3vTe">interval</a> 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 <a href="/facts/Real_line/6eq42xX5">real line</a> R, but the definitions go through <a href="/facts/Mutatis_mutandis/okly010I">mutatis mutandis</a> if S is Rn, a <a href="/facts/Normed_space/rmDljfyE">normed vector space</a>, or even a general <a href="/facts/Metric_space/gnBBRXtc">metric space</a>.

<h3>Continuity almost surely</h3>
Given a time t ∈ T, X is said to be continuous with probability one at t if

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    {\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.}

<h3>Mean-square continuity</h3>
Given a time t ∈ T, X is said to be continuous in mean-square at t if E[|Xt|2] < +∞ and

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

<h3>Continuity in probability</h3>
Main article: <a href="/facts/Continuity_in_probability/HKPlvgU8">Continuity in probability</a>
Given a time t ∈ T, X is said to be continuous in probability at t if, for all ε > 0,

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    {\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

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

<h3>Continuity in distribution</h3>
Given a time t ∈ T, X is said to be continuous in distribution at t if

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

for all points x at which Ft is continuous, where Ft denotes the <a href="/facts/Cumulative_distribution_function/WaKU8tp4">cumulative distribution function</a> of the <a href="/facts/Random_variable/TwTBXnLT">random variable</a> Xt.

<h3>Sample continuity</h3>
Main article: <a href="/facts/Sample_continuous_process/sWFG0MEJ">Sample continuous process</a>
X is said to be sample continuous if Xt(ω) is continuous in t for P-<a href="/facts/Almost_all/KvHxUCCc">almost all</a> ω ∈ Ω. Sample continuity is the appropriate notion of continuity for processes such as <a href="/facts/It%C5%8D_diffusion/vlwAqWSU">Itō diffusions</a>.

<h3>Feller continuity</h3>
Main article: <a href="/facts/Feller-continuous_process/pFckbcUT">Feller-continuous process</a>
X is said to be a Feller-continuous process if, for any fixed t ∈ T and any <a href="/facts/Bounded_function/whVjbtuk">bounded</a>, continuous and Σ-<a href="/facts/Measurable_function/NgIHnb1b">measurable function</a> g : S → R, Ex[g(Xt)] 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.

<h2 id="relationships">Relationships</h2>
The relationships between the various types of continuity of stochastic processes are akin to the relationships between the various types of <a href="/facts/Convergence_of_random_variables/pWdatFlY">convergence of random variables</a>. In particular:

<ul><li>continuity with probability one implies continuity in probability;</li>
<li>continuity in mean-square implies continuity in probability;</li>
<li>continuity with probability one neither implies, nor is implied by, continuity in mean-square;</li>
<li>continuity in probability implies, but is not implied by, continuity in distribution.</li></ul>
It is tempting to confuse continuity with probability one with sample continuity. Continuity with probability one at time t means that P(At) = 0, where the event At is given by

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    {\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

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

A is an <a href="/facts/Uncountable/zl2WqyJQ">uncountable</a> <a href="/facts/Union_(set_theory)/KVVXKfWl">union</a> 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(At) = 0 for every t ∈ T. This is the case, for example, with the <a href="/facts/Telegraph_process/0MYYdDuJ">telegraph process</a>.

<h2 id="notes">Notes</h2>

<ul><li>Kloeden, Peter E.; Platen, Eckhard (1992). Numerical solution of stochastic differential equations. Applications of Mathematics (New York) 23. Berlin: Springer-Verlag. pp. 38–39. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 3-540-54062-8.</li>
<li><a href="/facts/Bernt_%C3%98ksendal/Au4oSbnG">Øksendal, Bernt K.</a> (2003). Stochastic Differential Equations: An Introduction with Applications (Sixth ed.). Berlin: Springer. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 3-540-04758-1. (See Lemma 8.1.4)</li></ul>

<h2 id="references">References</h2>

<ol>
<li id="fn:1">Dodge, Y. (2006) The Oxford Dictionary of Statistical Terms, OUP. ISBN 0-19-920613-9 (Entry for "continuous process") <a href="/wiki/ISBN_(identifier)" target="_blank">/wiki/ISBN_(identifier)</a> <a href="#fnref:1" class="footnote-back-ref">↩</a></li>
<li id="fn:2">Dodge, Y. (2006) The Oxford Dictionary of Statistical Terms, OUP. ISBN 0-19-920613-9 (Entry for "continuous process") <a href="/wiki/ISBN_(identifier)" target="_blank">/wiki/ISBN_(identifier)</a> <a href="#fnref:2" class="footnote-back-ref">↩</a></li>
</ol>

Continuous stochastic process open-in-new

Continuous stochastic process