Is the following stopping time finite: $T:=inf{tgeq 0:B_tgeq sqrt{t}+1}?$
0
$begingroup$
We have a Brownian motion process $B$ and a stopping time defined like this:
$$T:=inf{tgeq 0:B_tgeq sqrt{t}+1}.$$
Is this stopping time almost surely finite, eg. $T<infty$, and why?
My intuition would say it is.
brownian-motion stopping-times
edited Dec 12 '18 at 23:50
zoidberg
1,080113
asked Dec 12 '18 at 23:47
RavonripRavonrip
898
$endgroup$
add a comment |
0
$begingroup$
We have a Brownian motion process $B$ and a stopping time defined like this:
$$T:=inf{tgeq 0:B_tgeq sqrt{t}+1}.$$
Is this stopping time almost surely finite, eg. $T<infty$, and why?
My intuition would say it is.
brownian-motion stopping-times
edited Dec 12 '18 at 23:50
zoidberg
1,080113
asked Dec 12 '18 at 23:47
RavonripRavonrip
898
$endgroup$
$begingroup$
What is your intuition?
$endgroup$
– zoidberg
Dec 12 '18 at 23:51
1
$begingroup$
That we know the $limsup_{tto infty}frac{B_t}{sqrt{t}}=infty$ a.s. And that means that the BM has to go over $sqrt{t}$ at some point.
$endgroup$
– Ravonrip
Dec 12 '18 at 23:56
1
$begingroup$
If you know that, then isn't that a proof? How do you know that limsup is $infty$?
$endgroup$
– zoidberg
Dec 13 '18 at 0:05
add a comment |
0
0
0
$begingroup$
We have a Brownian motion process $B$ and a stopping time defined like this:
$$T:=inf{tgeq 0:B_tgeq sqrt{t}+1}.$$
Is this stopping time almost surely finite, eg. $T<infty$, and why?
My intuition would say it is.
brownian-motion stopping-times
edited Dec 12 '18 at 23:50
zoidberg
1,080113
asked Dec 12 '18 at 23:47
RavonripRavonrip
898
$endgroup$
We have a Brownian motion process $B$ and a stopping time defined like this:
$$T:=inf{tgeq 0:B_tgeq sqrt{t}+1}.$$
Is this stopping time almost surely finite, eg. $T<infty$, and why?
My intuition would say it is.
brownian-motion stopping-times
brownian-motion stopping-times
edited Dec 12 '18 at 23:50
zoidberg
1,080113
asked Dec 12 '18 at 23:47
RavonripRavonrip
898
edited Dec 12 '18 at 23:50
zoidberg
1,080113
asked Dec 12 '18 at 23:47
RavonripRavonrip
898
edited Dec 12 '18 at 23:50
zoidberg
1,080113
edited Dec 12 '18 at 23:50
zoidberg
1,080113
edited Dec 12 '18 at 23:50
zoidberg
1,080113
1,080113
asked Dec 12 '18 at 23:47
RavonripRavonrip
898
asked Dec 12 '18 at 23:47
RavonripRavonrip
898
asked Dec 12 '18 at 23:47
RavonripRavonrip
898
898
$begingroup$
What is your intuition?
$endgroup$
– zoidberg
Dec 12 '18 at 23:51
1
$begingroup$
That we know the $limsup_{tto infty}frac{B_t}{sqrt{t}}=infty$ a.s. And that means that the BM has to go over $sqrt{t}$ at some point.
$endgroup$
– Ravonrip
Dec 12 '18 at 23:56
1
$begingroup$
If you know that, then isn't that a proof? How do you know that limsup is $infty$?
$endgroup$
– zoidberg
Dec 13 '18 at 0:05
add a comment |
$begingroup$
What is your intuition?
$endgroup$
– zoidberg
Dec 12 '18 at 23:51
1
$begingroup$
That we know the $limsup_{tto infty}frac{B_t}{sqrt{t}}=infty$ a.s. And that means that the BM has to go over $sqrt{t}$ at some point.
$endgroup$
– Ravonrip
Dec 12 '18 at 23:56
1
$begingroup$
If you know that, then isn't that a proof? How do you know that limsup is $infty$?
$endgroup$
– zoidberg
Dec 13 '18 at 0:05
$begingroup$
What is your intuition?
$endgroup$
– zoidberg
Dec 12 '18 at 23:51
$begingroup$
What is your intuition?
$endgroup$
– zoidberg
Dec 12 '18 at 23:51
1
1
$begingroup$
That we know the $limsup_{tto infty}frac{B_t}{sqrt{t}}=infty$ a.s. And that means that the BM has to go over $sqrt{t}$ at some point.
$endgroup$
– Ravonrip
Dec 12 '18 at 23:56
$begingroup$
That we know the $limsup_{tto infty}frac{B_t}{sqrt{t}}=infty$ a.s. And that means that the BM has to go over $sqrt{t}$ at some point.
$endgroup$
– Ravonrip
Dec 12 '18 at 23:56
1
1
$begingroup$
If you know that, then isn't that a proof? How do you know that limsup is $infty$?
$endgroup$
– zoidberg
Dec 13 '18 at 0:05
$begingroup$
If you know that, then isn't that a proof? How do you know that limsup is $infty$?
$endgroup$
– zoidberg
Dec 13 '18 at 0:05
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
});
}
});
draft saved
draft discarded
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%2f3037403%2fis-the-following-stopping-time-finite-t-inf-t-geq-0b-t-geq-sqrtt1%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
draft saved
draft discarded
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.
draft saved
draft discarded
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%2f3037403%2fis-the-following-stopping-time-finite-t-inf-t-geq-0b-t-geq-sqrtt1%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 your intuition?
$endgroup$
– zoidberg
Dec 12 '18 at 23:51
1
$begingroup$
That we know the $limsup_{tto infty}frac{B_t}{sqrt{t}}=infty$ a.s. And that means that the BM has to go over $sqrt{t}$ at some point.
$endgroup$
– Ravonrip
Dec 12 '18 at 23:56
1
$begingroup$
If you know that, then isn't that a proof? How do you know that limsup is $infty$?
$endgroup$
– zoidberg
Dec 13 '18 at 0:05