Mixing
square integrable martingale Hilbert space
I read that $mathcal{M}^2 = {X:X text{ is a cadlag, square integrable martingale}}$ is isomorphic to $L^2(Omega, mathcal{F}_{infty}, mathbb{P})$ as hilbert spaces if we identify martingales that are modifications (since these are cadlag they are also indistinguishable).
The mapping is $phi(M_t) = M_{infty}$. I'm not sure why this is surjective. more precisely, I'm not sure why defining $X_t = mathbb{E}[X_{infty}|mathcal{F}_t]$ yields a cadlag or right continuous process, could someone point out the reason or pass me a reference?
martingales stochastic-analysis
edited Nov 22 '18 at 0:21
Bernard
118k639112
asked Nov 21 '18 at 23:59
CeeersonCeeerson
625
add a comment |
I read that $mathcal{M}^2 = {X:X text{ is a cadlag, square integrable martingale}}$ is isomorphic to $L^2(Omega, mathcal{F}_{infty}, mathbb{P})$ as hilbert spaces if we identify martingales that are modifications (since these are cadlag they are also indistinguishable).
The mapping is $phi(M_t) = M_{infty}$. I'm not sure why this is surjective. more precisely, I'm not sure why defining $X_t = mathbb{E}[X_{infty}|mathcal{F}_t]$ yields a cadlag or right continuous process, could someone point out the reason or pass me a reference?
martingales stochastic-analysis
edited Nov 22 '18 at 0:21
Bernard
118k639112
asked Nov 21 '18 at 23:59
CeeersonCeeerson
625
Take a look at Section II.2 in Continuous martingales and Brownian motion by Revuz & Yor.
– saz
Nov 22 '18 at 20:29
add a comment |
I read that $mathcal{M}^2 = {X:X text{ is a cadlag, square integrable martingale}}$ is isomorphic to $L^2(Omega, mathcal{F}_{infty}, mathbb{P})$ as hilbert spaces if we identify martingales that are modifications (since these are cadlag they are also indistinguishable).
The mapping is $phi(M_t) = M_{infty}$. I'm not sure why this is surjective. more precisely, I'm not sure why defining $X_t = mathbb{E}[X_{infty}|mathcal{F}_t]$ yields a cadlag or right continuous process, could someone point out the reason or pass me a reference?
martingales stochastic-analysis
edited Nov 22 '18 at 0:21
Bernard
118k639112
asked Nov 21 '18 at 23:59
CeeersonCeeerson
625
I read that $mathcal{M}^2 = {X:X text{ is a cadlag, square integrable martingale}}$ is isomorphic to $L^2(Omega, mathcal{F}_{infty}, mathbb{P})$ as hilbert spaces if we identify martingales that are modifications (since these are cadlag they are also indistinguishable).
The mapping is $phi(M_t) = M_{infty}$. I'm not sure why this is surjective. more precisely, I'm not sure why defining $X_t = mathbb{E}[X_{infty}|mathcal{F}_t]$ yields a cadlag or right continuous process, could someone point out the reason or pass me a reference?
martingales stochastic-analysis
martingales stochastic-analysis
edited Nov 22 '18 at 0:21
Bernard
118k639112
asked Nov 21 '18 at 23:59
CeeersonCeeerson
625
edited Nov 22 '18 at 0:21
Bernard
118k639112
asked Nov 21 '18 at 23:59
CeeersonCeeerson
625
edited Nov 22 '18 at 0:21
Bernard
118k639112
edited Nov 22 '18 at 0:21
Bernard
118k639112
edited Nov 22 '18 at 0:21
Bernard
118k639112
118k639112
asked Nov 21 '18 at 23:59
CeeersonCeeerson
625
asked Nov 21 '18 at 23:59
CeeersonCeeerson
625
asked Nov 21 '18 at 23:59
CeeersonCeeerson
625
625
Take a look at Section II.2 in Continuous martingales and Brownian motion by Revuz & Yor.
– saz
Nov 22 '18 at 20:29
add a comment |
Take a look at Section II.2 in Continuous martingales and Brownian motion by Revuz & Yor.
– saz
Nov 22 '18 at 20:29
Take a look at Section II.2 in Continuous martingales and Brownian motion by Revuz & Yor.
– saz
Nov 22 '18 at 20:29
Take a look at Section II.2 in Continuous martingales and Brownian motion by Revuz & Yor.
– saz
Nov 22 '18 at 20:29
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%2f3008559%2fsquare-integrable-martingale-hilbert-space%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%2f3008559%2fsquare-integrable-martingale-hilbert-space%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
Take a look at Section II.2 in Continuous martingales and Brownian motion by Revuz & Yor.
– saz
Nov 22 '18 at 20:29