Variance of the stochastic process
$begingroup$
I am asked to find $Var(aW_t+bint_0^tf(s)W_sds)$. This is what I did until now:
begin{align*} &mathbb{E}left(a^2W_t^2+2abW_tint_0^tf(s)W_sds+b^2left[int_0^tf(s)W_s , dsright]^2 right) \&=a^2t+2abint_0^tf(s)mathbb{E}(W_t,W_s)ds+b^2mathbb{E}left(left[int_0^tf(s)W_s , ds right]^2 right) \ &=a^2t+2abint_0^tf(s)sds+b^2mathbb{E}left(left[int_0^tf(s)W_s , ds right]^2 right) end{align*}
but I do not know if I am doing it correctly and how to calculate the last expectation.
stochastic-processes stochastic-calculus variance
edited Dec 13 '18 at 18:31
saz
81.3k861127
asked Dec 13 '18 at 18:01
T.SokhT.Sokh
6
$endgroup$
add a comment |
$begingroup$
I am asked to find $Var(aW_t+bint_0^tf(s)W_sds)$. This is what I did until now:
begin{align*} &mathbb{E}left(a^2W_t^2+2abW_tint_0^tf(s)W_sds+b^2left[int_0^tf(s)W_s , dsright]^2 right) \&=a^2t+2abint_0^tf(s)mathbb{E}(W_t,W_s)ds+b^2mathbb{E}left(left[int_0^tf(s)W_s , ds right]^2 right) \ &=a^2t+2abint_0^tf(s)sds+b^2mathbb{E}left(left[int_0^tf(s)W_s , ds right]^2 right) end{align*}
but I do not know if I am doing it correctly and how to calculate the last expectation.
stochastic-processes stochastic-calculus variance
edited Dec 13 '18 at 18:31
saz
81.3k861127
asked Dec 13 '18 at 18:01
T.SokhT.Sokh
6
$endgroup$
1
$begingroup$
Write $$left( int_0^t f(s) W_s , ds right)^2 = int_0^t int_0^t f(s) W_s f(u) W_u , ds , du$$ and apply Fubini's theorem
$endgroup$
– saz
Dec 13 '18 at 18:33
$begingroup$
Why not? You just need to compute $mathbb{E}(W_s W_u)$
$endgroup$
– saz
Dec 13 '18 at 20:22
$begingroup$
I got it thank you!
$endgroup$
– T.Sokh
Dec 13 '18 at 20:25
add a comment |
$begingroup$
I am asked to find $Var(aW_t+bint_0^tf(s)W_sds)$. This is what I did until now:
begin{align*} &mathbb{E}left(a^2W_t^2+2abW_tint_0^tf(s)W_sds+b^2left[int_0^tf(s)W_s , dsright]^2 right) \&=a^2t+2abint_0^tf(s)mathbb{E}(W_t,W_s)ds+b^2mathbb{E}left(left[int_0^tf(s)W_s , ds right]^2 right) \ &=a^2t+2abint_0^tf(s)sds+b^2mathbb{E}left(left[int_0^tf(s)W_s , ds right]^2 right) end{align*}
but I do not know if I am doing it correctly and how to calculate the last expectation.
stochastic-processes stochastic-calculus variance
edited Dec 13 '18 at 18:31
saz
81.3k861127
asked Dec 13 '18 at 18:01
T.SokhT.Sokh
6
$endgroup$
I am asked to find $Var(aW_t+bint_0^tf(s)W_sds)$. This is what I did until now:
begin{align*} &mathbb{E}left(a^2W_t^2+2abW_tint_0^tf(s)W_sds+b^2left[int_0^tf(s)W_s , dsright]^2 right) \&=a^2t+2abint_0^tf(s)mathbb{E}(W_t,W_s)ds+b^2mathbb{E}left(left[int_0^tf(s)W_s , ds right]^2 right) \ &=a^2t+2abint_0^tf(s)sds+b^2mathbb{E}left(left[int_0^tf(s)W_s , ds right]^2 right) end{align*}
but I do not know if I am doing it correctly and how to calculate the last expectation.
stochastic-processes stochastic-calculus variance
stochastic-processes stochastic-calculus variance
edited Dec 13 '18 at 18:31
saz
81.3k861127
asked Dec 13 '18 at 18:01
T.SokhT.Sokh
6
edited Dec 13 '18 at 18:31
saz
81.3k861127
asked Dec 13 '18 at 18:01
T.SokhT.Sokh
6
edited Dec 13 '18 at 18:31
saz
81.3k861127
edited Dec 13 '18 at 18:31
saz
81.3k861127
edited Dec 13 '18 at 18:31
saz
81.3k861127
81.3k861127
asked Dec 13 '18 at 18:01
T.SokhT.Sokh
6
asked Dec 13 '18 at 18:01
T.SokhT.Sokh
6
asked Dec 13 '18 at 18:01
T.SokhT.Sokh
6
6
1
$begingroup$
Write $$left( int_0^t f(s) W_s , ds right)^2 = int_0^t int_0^t f(s) W_s f(u) W_u , ds , du$$ and apply Fubini's theorem
$endgroup$
– saz
Dec 13 '18 at 18:33
$begingroup$
Why not? You just need to compute $mathbb{E}(W_s W_u)$
$endgroup$
– saz
Dec 13 '18 at 20:22
$begingroup$
I got it thank you!
$endgroup$
– T.Sokh
Dec 13 '18 at 20:25
add a comment |
1
$begingroup$
Write $$left( int_0^t f(s) W_s , ds right)^2 = int_0^t int_0^t f(s) W_s f(u) W_u , ds , du$$ and apply Fubini's theorem
$endgroup$
– saz
Dec 13 '18 at 18:33
$begingroup$
Why not? You just need to compute $mathbb{E}(W_s W_u)$
$endgroup$
– saz
Dec 13 '18 at 20:22
$begingroup$
I got it thank you!
$endgroup$
– T.Sokh
Dec 13 '18 at 20:25
1
1
$begingroup$
Write $$left( int_0^t f(s) W_s , ds right)^2 = int_0^t int_0^t f(s) W_s f(u) W_u , ds , du$$ and apply Fubini's theorem
$endgroup$
– saz
Dec 13 '18 at 18:33
$begingroup$
Write $$left( int_0^t f(s) W_s , ds right)^2 = int_0^t int_0^t f(s) W_s f(u) W_u , ds , du$$ and apply Fubini's theorem
$endgroup$
– saz
Dec 13 '18 at 18:33
$begingroup$
Why not? You just need to compute $mathbb{E}(W_s W_u)$
$endgroup$
– saz
Dec 13 '18 at 20:22
$begingroup$
Why not? You just need to compute $mathbb{E}(W_s W_u)$
$endgroup$
– saz
Dec 13 '18 at 20:22
$begingroup$
I got it thank you!
$endgroup$
– T.Sokh
Dec 13 '18 at 20:25
$begingroup$
I got it thank you!
$endgroup$
– T.Sokh
Dec 13 '18 at 20:25
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%2f3038382%2fvariance-of-the-stochastic-process%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%2f3038382%2fvariance-of-the-stochastic-process%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
$begingroup$
Write $$left( int_0^t f(s) W_s , ds right)^2 = int_0^t int_0^t f(s) W_s f(u) W_u , ds , du$$ and apply Fubini's theorem
$endgroup$
– saz
Dec 13 '18 at 18:33
$begingroup$
Why not? You just need to compute $mathbb{E}(W_s W_u)$
$endgroup$
– saz
Dec 13 '18 at 20:22
$begingroup$
I got it thank you!
$endgroup$
– T.Sokh
Dec 13 '18 at 20:25