Confusion about notation on continuous pathed stochastic processes
$begingroup$
In the definition below, I am unsure on what the $[omega]$ in $X(t)[omega]$ is meant to signify.
Is it saying that we first observe a stochastic process, create the mapping with the seen values of the random variables, then we denote values of the random variables by $X(t)[omega]$?
stochastic-processes
asked Jan 1 at 23:57
DLBDLB
548
$endgroup$
add a comment |
$begingroup$
In the definition below, I am unsure on what the $[omega]$ in $X(t)[omega]$ is meant to signify.
Is it saying that we first observe a stochastic process, create the mapping with the seen values of the random variables, then we denote values of the random variables by $X(t)[omega]$?
stochastic-processes
asked Jan 1 at 23:57
DLBDLB
548
$endgroup$
add a comment |
$begingroup$
In the definition below, I am unsure on what the $[omega]$ in $X(t)[omega]$ is meant to signify.
Is it saying that we first observe a stochastic process, create the mapping with the seen values of the random variables, then we denote values of the random variables by $X(t)[omega]$?
stochastic-processes
asked Jan 1 at 23:57
DLBDLB
548
$endgroup$
In the definition below, I am unsure on what the $[omega]$ in $X(t)[omega]$ is meant to signify.
Is it saying that we first observe a stochastic process, create the mapping with the seen values of the random variables, then we denote values of the random variables by $X(t)[omega]$?
stochastic-processes
stochastic-processes
asked Jan 1 at 23:57
DLBDLB
548
asked Jan 1 at 23:57
DLBDLB
548
asked Jan 1 at 23:57
DLBDLB
548
asked Jan 1 at 23:57
DLBDLB
548
asked Jan 1 at 23:57
DLBDLB
548
548
add a comment |
add a comment |
1 Answer
1
active
oldest
votes
$begingroup$
Your stochastic process ${X(t),tgeq0}$ is defined on some underlying set $Omega$ (which we don't make reference to very often). This way, for each $tgeq0$, $X(t)$ is a function $$X(t):Omegatomathbb{R}.$$ $omega$ is just a generic element of the underlying space $Omega$. Thus for each $tgeq0$ and $omegainOmega$, $X(t)(omega)$ is a real number. By fixing $omegainOmega$ we can think of the mapping $$tmapsto X(t)(omega)$$ as a real-valued function over time.
The definition provided is just saying that the set of all $omegainOmega$ for which the above mapping is continuous is required to have probability measure 1.
answered Jan 2 at 7:44
user375366user375366
885138
$endgroup$
add a comment |
Your Answer
StackExchange.ifUsing("editor", function () {
return StackExchange.using("mathjaxEditing", function () {
StackExchange.MarkdownEditor.creationCallbacks.add(function (editor, postfix) {
StackExchange.mathjaxEditing.prepareWmdForMathJax(editor, postfix, [["$", "$"], ["\\(","\\)"]]);
});
});
}, "mathjax-editing");
StackExchange.ready(function() {
var channelOptions = {
tags: "".split(" "),
id: "69"
};
initTagRenderer("".split(" "), "".split(" "), channelOptions);
StackExchange.using("externalEditor", function() {
// Have to fire editor after snippets, if snippets enabled
if (StackExchange.settings.snippets.snippetsEnabled) {
StackExchange.using("snippets", function() {
createEditor();
});
}
else {
createEditor();
}
});
function createEditor() {
StackExchange.prepareEditor({
heartbeatType: 'answer',
autoActivateHeartbeat: false,
convertImagesToLinks: true,
noModals: true,
showLowRepImageUploadWarning: true,
reputationToPostImages: 10,
bindNavPrevention: true,
postfix: "",
imageUploader: {
brandingHtml: "Powered by u003ca class="icon-imgur-white" href="https://imgur.com/"u003eu003c/au003e",
contentPolicyHtml: "User contributions licensed under u003ca href="https://creativecommons.org/licenses/by-sa/3.0/"u003ecc by-sa 3.0 with attribution requiredu003c/au003e u003ca href="https://stackoverflow.com/legal/content-policy"u003e(content policy)u003c/au003e",
allowUrls: true
},
noCode: true, onDemand: true,
discardSelector: ".discard-answer"
,immediatelyShowMarkdownHelp:true
});
}
});
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
StackExchange.ready(
function () {
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f3059005%2fconfusion-about-notation-on-continuous-pathed-stochastic-processes%23new-answer', 'question_page');
}
);
Post as a guest
Required, but never shown
1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
$begingroup$
Your stochastic process ${X(t),tgeq0}$ is defined on some underlying set $Omega$ (which we don't make reference to very often). This way, for each $tgeq0$, $X(t)$ is a function $$X(t):Omegatomathbb{R}.$$ $omega$ is just a generic element of the underlying space $Omega$. Thus for each $tgeq0$ and $omegainOmega$, $X(t)(omega)$ is a real number. By fixing $omegainOmega$ we can think of the mapping $$tmapsto X(t)(omega)$$ as a real-valued function over time.
The definition provided is just saying that the set of all $omegainOmega$ for which the above mapping is continuous is required to have probability measure 1.
answered Jan 2 at 7:44
user375366user375366
885138
$endgroup$
add a comment |
$begingroup$
Your stochastic process ${X(t),tgeq0}$ is defined on some underlying set $Omega$ (which we don't make reference to very often). This way, for each $tgeq0$, $X(t)$ is a function $$X(t):Omegatomathbb{R}.$$ $omega$ is just a generic element of the underlying space $Omega$. Thus for each $tgeq0$ and $omegainOmega$, $X(t)(omega)$ is a real number. By fixing $omegainOmega$ we can think of the mapping $$tmapsto X(t)(omega)$$ as a real-valued function over time.
The definition provided is just saying that the set of all $omegainOmega$ for which the above mapping is continuous is required to have probability measure 1.
answered Jan 2 at 7:44
user375366user375366
885138
$endgroup$
add a comment |
$begingroup$
Your stochastic process ${X(t),tgeq0}$ is defined on some underlying set $Omega$ (which we don't make reference to very often). This way, for each $tgeq0$, $X(t)$ is a function $$X(t):Omegatomathbb{R}.$$ $omega$ is just a generic element of the underlying space $Omega$. Thus for each $tgeq0$ and $omegainOmega$, $X(t)(omega)$ is a real number. By fixing $omegainOmega$ we can think of the mapping $$tmapsto X(t)(omega)$$ as a real-valued function over time.
The definition provided is just saying that the set of all $omegainOmega$ for which the above mapping is continuous is required to have probability measure 1.
answered Jan 2 at 7:44
user375366user375366
885138
$endgroup$
Your stochastic process ${X(t),tgeq0}$ is defined on some underlying set $Omega$ (which we don't make reference to very often). This way, for each $tgeq0$, $X(t)$ is a function $$X(t):Omegatomathbb{R}.$$ $omega$ is just a generic element of the underlying space $Omega$. Thus for each $tgeq0$ and $omegainOmega$, $X(t)(omega)$ is a real number. By fixing $omegainOmega$ we can think of the mapping $$tmapsto X(t)(omega)$$ as a real-valued function over time.
The definition provided is just saying that the set of all $omegainOmega$ for which the above mapping is continuous is required to have probability measure 1.
answered Jan 2 at 7:44
user375366user375366
885138
answered Jan 2 at 7:44
user375366user375366
885138
answered Jan 2 at 7:44
user375366user375366
885138
answered Jan 2 at 7:44
user375366user375366
885138
885138
add a comment |
add a comment |
Thanks for contributing an answer to Mathematics Stack Exchange!
- Please be sure to answer the question. Provide details and share your research!
But avoid …
- Asking for help, clarification, or responding to other answers.
- Making statements based on opinion; back them up with references or personal experience.
Use MathJax to format equations. MathJax reference.
To learn more, see our tips on writing great answers.
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
StackExchange.ready(
function () {
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f3059005%2fconfusion-about-notation-on-continuous-pathed-stochastic-processes%23new-answer', 'question_page');
}
);
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
