Counterexample in Kolmogorov theorem about existence of almost surely continuous modification
I want to understand this Kolmogorov theorem about existence of almost surely continuous modification:
A process ${xi_t, in[0,T]}$ admits an almost surely continuous modification if there exist constants $a,b,c>0$ such that $$mathbb{E}[|xi_t-xi_s|^a]leq b|t-s|^{1+c}$$ for all $s,tin [0,T]$.
And I have such a question. What if we take a process ${xi_t=e^{w_t^3}, tin[0,T]}$, where $w_t$ is standard wiener process. Then will it be determined $mathbb{E}[|xi_t-xi_s|^a]$ ?
stochastic-processes examples-counterexamples expected-value
edited Dec 3 '18 at 15:08
GNUSupporter 8964民主女神 地下教會
12.8k72445
asked Dec 3 '18 at 14:58
EmeraldEmerald
378
|
show 1 more comment
I want to understand this Kolmogorov theorem about existence of almost surely continuous modification:
A process ${xi_t, in[0,T]}$ admits an almost surely continuous modification if there exist constants $a,b,c>0$ such that $$mathbb{E}[|xi_t-xi_s|^a]leq b|t-s|^{1+c}$$ for all $s,tin [0,T]$.
And I have such a question. What if we take a process ${xi_t=e^{w_t^3}, tin[0,T]}$, where $w_t$ is standard wiener process. Then will it be determined $mathbb{E}[|xi_t-xi_s|^a]$ ?
stochastic-processes examples-counterexamples expected-value
edited Dec 3 '18 at 15:08
GNUSupporter 8964民主女神 地下教會
12.8k72445
asked Dec 3 '18 at 14:58
EmeraldEmerald
378
1
You are right, this will not be integrable. However, the Kolmogorov theorem it also true with $|cdot|^a$ replaced by $max{|cdot|^a,1}$.
– zhoraster
Dec 3 '18 at 15:12
@zhoraster Thank you very much for your answer.
– Emerald
Dec 3 '18 at 15:14
@zhoraster The problem is that I still can't understand why we conclude about non-integrability. Why is it true $forall a$?
– Emerald
Dec 5 '18 at 17:25
The function $e^{a |x|^3 - b x^2}$ is non-integrable for any $a>0$, $binmathbb{R}$.
– zhoraster
Dec 5 '18 at 20:57
@zhoraster Sorry, but how did we move from this $e^{w_t^3}$ to this $e^{a|x|^3−bx^2}$ ?
– Emerald
Dec 10 '18 at 17:08
|
show 1 more comment
I want to understand this Kolmogorov theorem about existence of almost surely continuous modification:
A process ${xi_t, in[0,T]}$ admits an almost surely continuous modification if there exist constants $a,b,c>0$ such that $$mathbb{E}[|xi_t-xi_s|^a]leq b|t-s|^{1+c}$$ for all $s,tin [0,T]$.
And I have such a question. What if we take a process ${xi_t=e^{w_t^3}, tin[0,T]}$, where $w_t$ is standard wiener process. Then will it be determined $mathbb{E}[|xi_t-xi_s|^a]$ ?
stochastic-processes examples-counterexamples expected-value
edited Dec 3 '18 at 15:08
GNUSupporter 8964民主女神 地下教會
12.8k72445
asked Dec 3 '18 at 14:58
EmeraldEmerald
378
I want to understand this Kolmogorov theorem about existence of almost surely continuous modification:
A process ${xi_t, in[0,T]}$ admits an almost surely continuous modification if there exist constants $a,b,c>0$ such that $$mathbb{E}[|xi_t-xi_s|^a]leq b|t-s|^{1+c}$$ for all $s,tin [0,T]$.
And I have such a question. What if we take a process ${xi_t=e^{w_t^3}, tin[0,T]}$, where $w_t$ is standard wiener process. Then will it be determined $mathbb{E}[|xi_t-xi_s|^a]$ ?
stochastic-processes examples-counterexamples expected-value
stochastic-processes examples-counterexamples expected-value
edited Dec 3 '18 at 15:08
GNUSupporter 8964民主女神 地下教會
12.8k72445
asked Dec 3 '18 at 14:58
EmeraldEmerald
378
edited Dec 3 '18 at 15:08
GNUSupporter 8964民主女神 地下教會
12.8k72445
asked Dec 3 '18 at 14:58
EmeraldEmerald
378
edited Dec 3 '18 at 15:08
GNUSupporter 8964民主女神 地下教會
12.8k72445
edited Dec 3 '18 at 15:08
GNUSupporter 8964民主女神 地下教會
12.8k72445
edited Dec 3 '18 at 15:08
GNUSupporter 8964民主女神 地下教會
12.8k72445
12.8k72445
asked Dec 3 '18 at 14:58
EmeraldEmerald
378
asked Dec 3 '18 at 14:58
EmeraldEmerald
378
asked Dec 3 '18 at 14:58
EmeraldEmerald
378
378
1
You are right, this will not be integrable. However, the Kolmogorov theorem it also true with $|cdot|^a$ replaced by $max{|cdot|^a,1}$.
– zhoraster
Dec 3 '18 at 15:12
@zhoraster Thank you very much for your answer.
– Emerald
Dec 3 '18 at 15:14
@zhoraster The problem is that I still can't understand why we conclude about non-integrability. Why is it true $forall a$?
– Emerald
Dec 5 '18 at 17:25
The function $e^{a |x|^3 - b x^2}$ is non-integrable for any $a>0$, $binmathbb{R}$.
– zhoraster
Dec 5 '18 at 20:57
@zhoraster Sorry, but how did we move from this $e^{w_t^3}$ to this $e^{a|x|^3−bx^2}$ ?
– Emerald
Dec 10 '18 at 17:08
|
show 1 more comment
1
You are right, this will not be integrable. However, the Kolmogorov theorem it also true with $|cdot|^a$ replaced by $max{|cdot|^a,1}$.
– zhoraster
Dec 3 '18 at 15:12
@zhoraster Thank you very much for your answer.
– Emerald
Dec 3 '18 at 15:14
@zhoraster The problem is that I still can't understand why we conclude about non-integrability. Why is it true $forall a$?
– Emerald
Dec 5 '18 at 17:25
The function $e^{a |x|^3 - b x^2}$ is non-integrable for any $a>0$, $binmathbb{R}$.
– zhoraster
Dec 5 '18 at 20:57
@zhoraster Sorry, but how did we move from this $e^{w_t^3}$ to this $e^{a|x|^3−bx^2}$ ?
– Emerald
Dec 10 '18 at 17:08
1
1
You are right, this will not be integrable. However, the Kolmogorov theorem it also true with $|cdot|^a$ replaced by $max{|cdot|^a,1}$.
– zhoraster
Dec 3 '18 at 15:12
You are right, this will not be integrable. However, the Kolmogorov theorem it also true with $|cdot|^a$ replaced by $max{|cdot|^a,1}$.
– zhoraster
Dec 3 '18 at 15:12
@zhoraster Thank you very much for your answer.
– Emerald
Dec 3 '18 at 15:14
@zhoraster Thank you very much for your answer.
– Emerald
Dec 3 '18 at 15:14
@zhoraster The problem is that I still can't understand why we conclude about non-integrability. Why is it true $forall a$?
– Emerald
Dec 5 '18 at 17:25
@zhoraster The problem is that I still can't understand why we conclude about non-integrability. Why is it true $forall a$?
– Emerald
Dec 5 '18 at 17:25
The function $e^{a |x|^3 - b x^2}$ is non-integrable for any $a>0$, $binmathbb{R}$.
– zhoraster
Dec 5 '18 at 20:57
The function $e^{a |x|^3 - b x^2}$ is non-integrable for any $a>0$, $binmathbb{R}$.
– zhoraster
Dec 5 '18 at 20:57
@zhoraster Sorry, but how did we move from this $e^{w_t^3}$ to this $e^{a|x|^3−bx^2}$ ?
– Emerald
Dec 10 '18 at 17:08
@zhoraster Sorry, but how did we move from this $e^{w_t^3}$ to this $e^{a|x|^3−bx^2}$ ?
– Emerald
Dec 10 '18 at 17:08
|
show 1 more 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%2f3024165%2fcounterexample-in-kolmogorov-theorem-about-existence-of-almost-surely-continuous%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.
Some of your past answers have not been well-received, and you're in danger of being blocked from answering.
Please pay close attention to the following guidance:
- 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.
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%2f3024165%2fcounterexample-in-kolmogorov-theorem-about-existence-of-almost-surely-continuous%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
1
You are right, this will not be integrable. However, the Kolmogorov theorem it also true with $|cdot|^a$ replaced by $max{|cdot|^a,1}$.
– zhoraster
Dec 3 '18 at 15:12
@zhoraster Thank you very much for your answer.
– Emerald
Dec 3 '18 at 15:14
@zhoraster The problem is that I still can't understand why we conclude about non-integrability. Why is it true $forall a$?
– Emerald
Dec 5 '18 at 17:25
The function $e^{a |x|^3 - b x^2}$ is non-integrable for any $a>0$, $binmathbb{R}$.
– zhoraster
Dec 5 '18 at 20:57
@zhoraster Sorry, but how did we move from this $e^{w_t^3}$ to this $e^{a|x|^3−bx^2}$ ?
– Emerald
Dec 10 '18 at 17:08