Don't fully understand Theorem $5.13$ from Durrett
$begingroup$
Can anyone shed some light on this theorem? Not sure of the significance of the result or how one would apply it.
If $M_n$ is a supermartingale with respect to $X_n$ and $T$ is a stopping time then the stopped process $M_{T wedge n}$ is a supermartingale with respect to $X_n$. In particular, $mathrm EM_{T wedge n} leq M_0$.
stochastic-processes martingales
edited Dec 2 '18 at 4:43
Rócherz
2,7762721
asked Dec 2 '18 at 4:33
boatboat
61
$endgroup$
add a comment |
$begingroup$
Can anyone shed some light on this theorem? Not sure of the significance of the result or how one would apply it.
If $M_n$ is a supermartingale with respect to $X_n$ and $T$ is a stopping time then the stopped process $M_{T wedge n}$ is a supermartingale with respect to $X_n$. In particular, $mathrm EM_{T wedge n} leq M_0$.
stochastic-processes martingales
edited Dec 2 '18 at 4:43
Rócherz
2,7762721
asked Dec 2 '18 at 4:33
boatboat
61
$endgroup$
1
$begingroup$
Just search for "optional stopping theorem" or "optional sampling theorem" and you will find plenty of possible ways to apply it.
$endgroup$
– saz
Dec 2 '18 at 10:06
add a comment |
$begingroup$
Can anyone shed some light on this theorem? Not sure of the significance of the result or how one would apply it.
If $M_n$ is a supermartingale with respect to $X_n$ and $T$ is a stopping time then the stopped process $M_{T wedge n}$ is a supermartingale with respect to $X_n$. In particular, $mathrm EM_{T wedge n} leq M_0$.
stochastic-processes martingales
edited Dec 2 '18 at 4:43
Rócherz
2,7762721
asked Dec 2 '18 at 4:33
boatboat
61
$endgroup$
Can anyone shed some light on this theorem? Not sure of the significance of the result or how one would apply it.
If $M_n$ is a supermartingale with respect to $X_n$ and $T$ is a stopping time then the stopped process $M_{T wedge n}$ is a supermartingale with respect to $X_n$. In particular, $mathrm EM_{T wedge n} leq M_0$.
stochastic-processes martingales
stochastic-processes martingales
edited Dec 2 '18 at 4:43
Rócherz
2,7762721
asked Dec 2 '18 at 4:33
boatboat
61
edited Dec 2 '18 at 4:43
Rócherz
2,7762721
asked Dec 2 '18 at 4:33
boatboat
61
edited Dec 2 '18 at 4:43
Rócherz
2,7762721
edited Dec 2 '18 at 4:43
Rócherz
2,7762721
edited Dec 2 '18 at 4:43
Rócherz
2,7762721
2,7762721
asked Dec 2 '18 at 4:33
boatboat
61
asked Dec 2 '18 at 4:33
boatboat
61
asked Dec 2 '18 at 4:33
boatboat
61
61
1
$begingroup$
Just search for "optional stopping theorem" or "optional sampling theorem" and you will find plenty of possible ways to apply it.
$endgroup$
– saz
Dec 2 '18 at 10:06
add a comment |
1
$begingroup$
Just search for "optional stopping theorem" or "optional sampling theorem" and you will find plenty of possible ways to apply it.
$endgroup$
– saz
Dec 2 '18 at 10:06
1
1
$begingroup$
Just search for "optional stopping theorem" or "optional sampling theorem" and you will find plenty of possible ways to apply it.
$endgroup$
– saz
Dec 2 '18 at 10:06
$begingroup$
Just search for "optional stopping theorem" or "optional sampling theorem" and you will find plenty of possible ways to apply it.
$endgroup$
– saz
Dec 2 '18 at 10:06
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%2f3022235%2fdont-fully-understand-theorem-5-13-from-durrett%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%2f3022235%2fdont-fully-understand-theorem-5-13-from-durrett%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
$begingroup$
Just search for "optional stopping theorem" or "optional sampling theorem" and you will find plenty of possible ways to apply it.
$endgroup$
– saz
Dec 2 '18 at 10:06