The property of coercivity in stochastic analysis
$begingroup$
Given an SDE
$$ dX_{t}=b(t,X_{t})dt+sigma (t,X_{t}) dW_{t} $$
With $W$ the Wiener process. I have seen some results that under some assumptions on the coefficients $b,sigma$ such as :
i) Lipshitz
ii) Linear growth
iii) $sigma$ bounded
iv) $b$ coercive that is $b(t,X_{t})cdot X_{t} leq -alpha |X_{t}|^{2} $ where $alpha>0$
Then using krylov bogoliubov theorem we can find an invariant measure. My question is regarding the intuition behind the coercivity condition, I can see that the action of $dW_{t}$ is not really 'invariant' since it is a b.m and hence has Guassian distribution, therefore we need the drift term i.e the $dt$ to counteract the non-invarience of the b.m? What is it about coercivity that does this?
stochastic-calculus stochastic-analysis sde stationary-processes coercive
asked Dec 9 '18 at 14:09
MontyMonty
34113
$endgroup$
add a comment |
$begingroup$
Given an SDE
$$ dX_{t}=b(t,X_{t})dt+sigma (t,X_{t}) dW_{t} $$
With $W$ the Wiener process. I have seen some results that under some assumptions on the coefficients $b,sigma$ such as :
i) Lipshitz
ii) Linear growth
iii) $sigma$ bounded
iv) $b$ coercive that is $b(t,X_{t})cdot X_{t} leq -alpha |X_{t}|^{2} $ where $alpha>0$
Then using krylov bogoliubov theorem we can find an invariant measure. My question is regarding the intuition behind the coercivity condition, I can see that the action of $dW_{t}$ is not really 'invariant' since it is a b.m and hence has Guassian distribution, therefore we need the drift term i.e the $dt$ to counteract the non-invarience of the b.m? What is it about coercivity that does this?
stochastic-calculus stochastic-analysis sde stationary-processes coercive
asked Dec 9 '18 at 14:09
MontyMonty
34113
$endgroup$
add a comment |
$begingroup$
Given an SDE
$$ dX_{t}=b(t,X_{t})dt+sigma (t,X_{t}) dW_{t} $$
With $W$ the Wiener process. I have seen some results that under some assumptions on the coefficients $b,sigma$ such as :
i) Lipshitz
ii) Linear growth
iii) $sigma$ bounded
iv) $b$ coercive that is $b(t,X_{t})cdot X_{t} leq -alpha |X_{t}|^{2} $ where $alpha>0$
Then using krylov bogoliubov theorem we can find an invariant measure. My question is regarding the intuition behind the coercivity condition, I can see that the action of $dW_{t}$ is not really 'invariant' since it is a b.m and hence has Guassian distribution, therefore we need the drift term i.e the $dt$ to counteract the non-invarience of the b.m? What is it about coercivity that does this?
stochastic-calculus stochastic-analysis sde stationary-processes coercive
asked Dec 9 '18 at 14:09
MontyMonty
34113
$endgroup$
Given an SDE
$$ dX_{t}=b(t,X_{t})dt+sigma (t,X_{t}) dW_{t} $$
With $W$ the Wiener process. I have seen some results that under some assumptions on the coefficients $b,sigma$ such as :
i) Lipshitz
ii) Linear growth
iii) $sigma$ bounded
iv) $b$ coercive that is $b(t,X_{t})cdot X_{t} leq -alpha |X_{t}|^{2} $ where $alpha>0$
Then using krylov bogoliubov theorem we can find an invariant measure. My question is regarding the intuition behind the coercivity condition, I can see that the action of $dW_{t}$ is not really 'invariant' since it is a b.m and hence has Guassian distribution, therefore we need the drift term i.e the $dt$ to counteract the non-invarience of the b.m? What is it about coercivity that does this?
stochastic-calculus stochastic-analysis sde stationary-processes coercive
stochastic-calculus stochastic-analysis sde stationary-processes coercive
asked Dec 9 '18 at 14:09
MontyMonty
34113
asked Dec 9 '18 at 14:09
MontyMonty
34113
edited Dec 11 '18 at 18:05
Monty
asked Dec 9 '18 at 14:09
MontyMonty
34113
asked Dec 9 '18 at 14:09
MontyMonty
34113
asked Dec 9 '18 at 14:09
MontyMonty
34113
34113
add a comment |
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%2f3032409%2fthe-property-of-coercivity-in-stochastic-analysis%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%2f3032409%2fthe-property-of-coercivity-in-stochastic-analysis%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
