An example of the fact that from measurability of a random process does not follow measurability of its...
$begingroup$
Let {$ xi _t(omega), tin[0,infty)$} be a random process and $ xi _t(omega)in {mathfrak F_t}$ (some filtration). If $ xi _t(omega) $ is $ mathfrak F_t $ measurable then $int_0^txi _s(omega)L(ds)$ not necessarily $ mathfrak F_t $ measurable. I am looking for a process, that will demonstrate this statement. Thanks in advance for any tips.
measure-theory stochastic-processes lebesgue-integral examples-counterexamples filtrations
asked Dec 1 '18 at 14:36
EmeraldEmerald
378
$endgroup$
add a comment |
$begingroup$
Let {$ xi _t(omega), tin[0,infty)$} be a random process and $ xi _t(omega)in {mathfrak F_t}$ (some filtration). If $ xi _t(omega) $ is $ mathfrak F_t $ measurable then $int_0^txi _s(omega)L(ds)$ not necessarily $ mathfrak F_t $ measurable. I am looking for a process, that will demonstrate this statement. Thanks in advance for any tips.
measure-theory stochastic-processes lebesgue-integral examples-counterexamples filtrations
asked Dec 1 '18 at 14:36
EmeraldEmerald
378
$endgroup$
$begingroup$
What is $L$...? And what exactly do you mean by $xi_t(omega)$ is measurable (measurable with respect to which variable? $omega$ or $t$? or jointly measurable)?
$endgroup$
– saz
Dec 1 '18 at 17:31
$begingroup$
@saz Lebesgue measure
$endgroup$
– Emerald
Dec 1 '18 at 17:32
$begingroup$
I see. What makes you believe that such a process exist?
$endgroup$
– saz
Dec 1 '18 at 17:42
$begingroup$
@saz I think such an example could be Heaviside function. But I'm not sure.
$endgroup$
– Emerald
Dec 1 '18 at 17:45
add a comment |
$begingroup$
Let {$ xi _t(omega), tin[0,infty)$} be a random process and $ xi _t(omega)in {mathfrak F_t}$ (some filtration). If $ xi _t(omega) $ is $ mathfrak F_t $ measurable then $int_0^txi _s(omega)L(ds)$ not necessarily $ mathfrak F_t $ measurable. I am looking for a process, that will demonstrate this statement. Thanks in advance for any tips.
measure-theory stochastic-processes lebesgue-integral examples-counterexamples filtrations
asked Dec 1 '18 at 14:36
EmeraldEmerald
378
$endgroup$
Let {$ xi _t(omega), tin[0,infty)$} be a random process and $ xi _t(omega)in {mathfrak F_t}$ (some filtration). If $ xi _t(omega) $ is $ mathfrak F_t $ measurable then $int_0^txi _s(omega)L(ds)$ not necessarily $ mathfrak F_t $ measurable. I am looking for a process, that will demonstrate this statement. Thanks in advance for any tips.
measure-theory stochastic-processes lebesgue-integral examples-counterexamples filtrations
measure-theory stochastic-processes lebesgue-integral examples-counterexamples filtrations
asked Dec 1 '18 at 14:36
EmeraldEmerald
378
asked Dec 1 '18 at 14:36
EmeraldEmerald
378
edited Dec 1 '18 at 18:03
Emerald
asked Dec 1 '18 at 14:36
EmeraldEmerald
378
asked Dec 1 '18 at 14:36
EmeraldEmerald
378
asked Dec 1 '18 at 14:36
EmeraldEmerald
378
378
$begingroup$
What is $L$...? And what exactly do you mean by $xi_t(omega)$ is measurable (measurable with respect to which variable? $omega$ or $t$? or jointly measurable)?
$endgroup$
– saz
Dec 1 '18 at 17:31
$begingroup$
@saz Lebesgue measure
$endgroup$
– Emerald
Dec 1 '18 at 17:32
$begingroup$
I see. What makes you believe that such a process exist?
$endgroup$
– saz
Dec 1 '18 at 17:42
$begingroup$
@saz I think such an example could be Heaviside function. But I'm not sure.
$endgroup$
– Emerald
Dec 1 '18 at 17:45
add a comment |
$begingroup$
What is $L$...? And what exactly do you mean by $xi_t(omega)$ is measurable (measurable with respect to which variable? $omega$ or $t$? or jointly measurable)?
$endgroup$
– saz
Dec 1 '18 at 17:31
$begingroup$
@saz Lebesgue measure
$endgroup$
– Emerald
Dec 1 '18 at 17:32
$begingroup$
I see. What makes you believe that such a process exist?
$endgroup$
– saz
Dec 1 '18 at 17:42
$begingroup$
@saz I think such an example could be Heaviside function. But I'm not sure.
$endgroup$
– Emerald
Dec 1 '18 at 17:45
$begingroup$
What is $L$...? And what exactly do you mean by $xi_t(omega)$ is measurable (measurable with respect to which variable? $omega$ or $t$? or jointly measurable)?
$endgroup$
– saz
Dec 1 '18 at 17:31
$begingroup$
What is $L$...? And what exactly do you mean by $xi_t(omega)$ is measurable (measurable with respect to which variable? $omega$ or $t$? or jointly measurable)?
$endgroup$
– saz
Dec 1 '18 at 17:31
$begingroup$
@saz Lebesgue measure
$endgroup$
– Emerald
Dec 1 '18 at 17:32
$begingroup$
@saz Lebesgue measure
$endgroup$
– Emerald
Dec 1 '18 at 17:32
$begingroup$
I see. What makes you believe that such a process exist?
$endgroup$
– saz
Dec 1 '18 at 17:42
$begingroup$
I see. What makes you believe that such a process exist?
$endgroup$
– saz
Dec 1 '18 at 17:42
$begingroup$
@saz I think such an example could be Heaviside function. But I'm not sure.
$endgroup$
– Emerald
Dec 1 '18 at 17:45
$begingroup$
@saz I think such an example could be Heaviside function. But I'm not sure.
$endgroup$
– Emerald
Dec 1 '18 at 17:45
add a comment |
0
active
oldest
votes
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%2f3021418%2fan-example-of-the-fact-that-from-measurability-of-a-random-process-does-not-foll%23new-answer', 'question_page');
}
);
Post as a guest
Required, but never shown
0
active
oldest
votes
0
active
oldest
votes
active
oldest
votes
active
oldest
votes
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%2f3021418%2fan-example-of-the-fact-that-from-measurability-of-a-random-process-does-not-foll%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
$begingroup$
What is $L$...? And what exactly do you mean by $xi_t(omega)$ is measurable (measurable with respect to which variable? $omega$ or $t$? or jointly measurable)?
$endgroup$
– saz
Dec 1 '18 at 17:31
$begingroup$
@saz Lebesgue measure
$endgroup$
– Emerald
Dec 1 '18 at 17:32
$begingroup$
I see. What makes you believe that such a process exist?
$endgroup$
– saz
Dec 1 '18 at 17:42
$begingroup$
@saz I think such an example could be Heaviside function. But I'm not sure.
$endgroup$
– Emerald
Dec 1 '18 at 17:45