Meaning of $L^2 (mathcal{F}_T, P)$?
$begingroup$
The space $L^2(mathcal{F}_T, P)$ is used in my textbook, but does not seem to be defined anywhere that I can find (and is ommited in the symbols table at the end of the book). Here, $P$ is a probability measure, and ${mathcal{F}_t}$ is a filtration.
As I understand it, a function $f in L^2 (P)$ if $(int f^2 dP)^{1/2} < infty$
The notation $L^2(mathcal{F}_T, P)$ is used in the context of Ito integrals from $S < t < T$ where $mathcal{F}_t$ is the filtration generated by the Brownian motion process.
So what does $L^2(mathcal{F}_T, P)$ mean?
I am guessing it means that $int f^2 dP < infty$ and $f(t,omega)$ is $mathcal{F}_t$-adapted. But it's never stated explicitly.
stochastic-processes definition self-learning
edited Dec 15 '18 at 11:42
Andrews
3571317
asked Dec 15 '18 at 3:51
XiaomiXiaomi
1,050115
$endgroup$
add a comment |
$begingroup$
The space $L^2(mathcal{F}_T, P)$ is used in my textbook, but does not seem to be defined anywhere that I can find (and is ommited in the symbols table at the end of the book). Here, $P$ is a probability measure, and ${mathcal{F}_t}$ is a filtration.
As I understand it, a function $f in L^2 (P)$ if $(int f^2 dP)^{1/2} < infty$
The notation $L^2(mathcal{F}_T, P)$ is used in the context of Ito integrals from $S < t < T$ where $mathcal{F}_t$ is the filtration generated by the Brownian motion process.
So what does $L^2(mathcal{F}_T, P)$ mean?
I am guessing it means that $int f^2 dP < infty$ and $f(t,omega)$ is $mathcal{F}_t$-adapted. But it's never stated explicitly.
stochastic-processes definition self-learning
edited Dec 15 '18 at 11:42
Andrews
3571317
asked Dec 15 '18 at 3:51
XiaomiXiaomi
1,050115
$endgroup$
1
$begingroup$
Generally, the notation $L^p(X , Sigma , mu)$ is the space of $p$ integrable functions ($int |f|^p mathrm{d}mu < infty$) (up to equivalence of course), with $sigma$ algebra $Sigma$ and measure $mu$ on a measure space $X$ . So I think your guess is correct.
$endgroup$
– rubikscube09
Dec 15 '18 at 4:07
add a comment |
$begingroup$
The space $L^2(mathcal{F}_T, P)$ is used in my textbook, but does not seem to be defined anywhere that I can find (and is ommited in the symbols table at the end of the book). Here, $P$ is a probability measure, and ${mathcal{F}_t}$ is a filtration.
As I understand it, a function $f in L^2 (P)$ if $(int f^2 dP)^{1/2} < infty$
The notation $L^2(mathcal{F}_T, P)$ is used in the context of Ito integrals from $S < t < T$ where $mathcal{F}_t$ is the filtration generated by the Brownian motion process.
So what does $L^2(mathcal{F}_T, P)$ mean?
I am guessing it means that $int f^2 dP < infty$ and $f(t,omega)$ is $mathcal{F}_t$-adapted. But it's never stated explicitly.
stochastic-processes definition self-learning
edited Dec 15 '18 at 11:42
Andrews
3571317
asked Dec 15 '18 at 3:51
XiaomiXiaomi
1,050115
$endgroup$
The space $L^2(mathcal{F}_T, P)$ is used in my textbook, but does not seem to be defined anywhere that I can find (and is ommited in the symbols table at the end of the book). Here, $P$ is a probability measure, and ${mathcal{F}_t}$ is a filtration.
As I understand it, a function $f in L^2 (P)$ if $(int f^2 dP)^{1/2} < infty$
The notation $L^2(mathcal{F}_T, P)$ is used in the context of Ito integrals from $S < t < T$ where $mathcal{F}_t$ is the filtration generated by the Brownian motion process.
So what does $L^2(mathcal{F}_T, P)$ mean?
I am guessing it means that $int f^2 dP < infty$ and $f(t,omega)$ is $mathcal{F}_t$-adapted. But it's never stated explicitly.
stochastic-processes definition self-learning
stochastic-processes definition self-learning
edited Dec 15 '18 at 11:42
Andrews
3571317
asked Dec 15 '18 at 3:51
XiaomiXiaomi
1,050115
edited Dec 15 '18 at 11:42
Andrews
3571317
asked Dec 15 '18 at 3:51
XiaomiXiaomi
1,050115
edited Dec 15 '18 at 11:42
Andrews
3571317
edited Dec 15 '18 at 11:42
Andrews
3571317
edited Dec 15 '18 at 11:42
Andrews
3571317
3571317
asked Dec 15 '18 at 3:51
XiaomiXiaomi
1,050115
asked Dec 15 '18 at 3:51
XiaomiXiaomi
1,050115
asked Dec 15 '18 at 3:51
XiaomiXiaomi
1,050115
1,050115
1
$begingroup$
Generally, the notation $L^p(X , Sigma , mu)$ is the space of $p$ integrable functions ($int |f|^p mathrm{d}mu < infty$) (up to equivalence of course), with $sigma$ algebra $Sigma$ and measure $mu$ on a measure space $X$ . So I think your guess is correct.
$endgroup$
– rubikscube09
Dec 15 '18 at 4:07
add a comment |
1
$begingroup$
Generally, the notation $L^p(X , Sigma , mu)$ is the space of $p$ integrable functions ($int |f|^p mathrm{d}mu < infty$) (up to equivalence of course), with $sigma$ algebra $Sigma$ and measure $mu$ on a measure space $X$ . So I think your guess is correct.
$endgroup$
– rubikscube09
Dec 15 '18 at 4:07
1
1
$begingroup$
Generally, the notation $L^p(X , Sigma , mu)$ is the space of $p$ integrable functions ($int |f|^p mathrm{d}mu < infty$) (up to equivalence of course), with $sigma$ algebra $Sigma$ and measure $mu$ on a measure space $X$ . So I think your guess is correct.
$endgroup$
– rubikscube09
Dec 15 '18 at 4:07
$begingroup$
Generally, the notation $L^p(X , Sigma , mu)$ is the space of $p$ integrable functions ($int |f|^p mathrm{d}mu < infty$) (up to equivalence of course), with $sigma$ algebra $Sigma$ and measure $mu$ on a measure space $X$ . So I think your guess is correct.
$endgroup$
– rubikscube09
Dec 15 '18 at 4:07
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%2f3040148%2fmeaning-of-l2-mathcalf-t-p%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%2f3040148%2fmeaning-of-l2-mathcalf-t-p%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
$begingroup$
Generally, the notation $L^p(X , Sigma , mu)$ is the space of $p$ integrable functions ($int |f|^p mathrm{d}mu < infty$) (up to equivalence of course), with $sigma$ algebra $Sigma$ and measure $mu$ on a measure space $X$ . So I think your guess is correct.
$endgroup$
– rubikscube09
Dec 15 '18 at 4:07