Convergence of martingale (reference)
$begingroup$
I have a sequence of processes ${X^N(t)}_{tin [0,T]}$, $Ninmathbb N$ such that
$X^N(t)=x+M^N(t)$,
where $M^N(t)$ is a martingale with expectation $0$ and with quadratic variation $langle M^N rangle (t)$ such that
$langle M^N rangle(t)-int_0^tX^N(s)(1-X^N(s))dsxrightarrow[Nto+infty]{}0$ in probability.
I know that the process $X^N(t)$ converges (I proved its tightness), I would like to conclude that $X^N(t)to X(t)$ where $X(t)$ is the solution of the following system of SDE
$$dX(t)=sqrt{X(t)(1-X(t)}dB(t)$$
$$X(0)=x$$
with $B(t)$ the classic Brownian motion.
Could someone suggest me a book in which I can find a theorem that could help me with the convergence of martingales?
probability-theory reference-request stochastic-processes martingales sde
$endgroup$
add a comment |
$begingroup$
I have a sequence of processes ${X^N(t)}_{tin [0,T]}$, $Ninmathbb N$ such that
$X^N(t)=x+M^N(t)$,
where $M^N(t)$ is a martingale with expectation $0$ and with quadratic variation $langle M^N rangle (t)$ such that
$langle M^N rangle(t)-int_0^tX^N(s)(1-X^N(s))dsxrightarrow[Nto+infty]{}0$ in probability.
I know that the process $X^N(t)$ converges (I proved its tightness), I would like to conclude that $X^N(t)to X(t)$ where $X(t)$ is the solution of the following system of SDE
$$dX(t)=sqrt{X(t)(1-X(t)}dB(t)$$
$$X(0)=x$$
with $B(t)$ the classic Brownian motion.
Could someone suggest me a book in which I can find a theorem that could help me with the convergence of martingales?
probability-theory reference-request stochastic-processes martingales sde
$endgroup$
add a comment |
$begingroup$
I have a sequence of processes ${X^N(t)}_{tin [0,T]}$, $Ninmathbb N$ such that
$X^N(t)=x+M^N(t)$,
where $M^N(t)$ is a martingale with expectation $0$ and with quadratic variation $langle M^N rangle (t)$ such that
$langle M^N rangle(t)-int_0^tX^N(s)(1-X^N(s))dsxrightarrow[Nto+infty]{}0$ in probability.
I know that the process $X^N(t)$ converges (I proved its tightness), I would like to conclude that $X^N(t)to X(t)$ where $X(t)$ is the solution of the following system of SDE
$$dX(t)=sqrt{X(t)(1-X(t)}dB(t)$$
$$X(0)=x$$
with $B(t)$ the classic Brownian motion.
Could someone suggest me a book in which I can find a theorem that could help me with the convergence of martingales?
probability-theory reference-request stochastic-processes martingales sde
$endgroup$
I have a sequence of processes ${X^N(t)}_{tin [0,T]}$, $Ninmathbb N$ such that
$X^N(t)=x+M^N(t)$,
where $M^N(t)$ is a martingale with expectation $0$ and with quadratic variation $langle M^N rangle (t)$ such that
$langle M^N rangle(t)-int_0^tX^N(s)(1-X^N(s))dsxrightarrow[Nto+infty]{}0$ in probability.
I know that the process $X^N(t)$ converges (I proved its tightness), I would like to conclude that $X^N(t)to X(t)$ where $X(t)$ is the solution of the following system of SDE
$$dX(t)=sqrt{X(t)(1-X(t)}dB(t)$$
$$X(0)=x$$
with $B(t)$ the classic Brownian motion.
Could someone suggest me a book in which I can find a theorem that could help me with the convergence of martingales?
probability-theory reference-request stochastic-processes martingales sde
probability-theory reference-request stochastic-processes martingales sde
edited Jan 29 at 17:06


dafinguzman
2,69911126
2,69911126
asked Jan 28 at 15:20
user268193user268193
718
718
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%2f3090996%2fconvergence-of-martingale-reference%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%2f3090996%2fconvergence-of-martingale-reference%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