Expectation in a stochastic differential equation The Next CEO of Stack OverflowWhat is Ito's lemma used for in quantitative finance?Question about the stochastic differential equation in the Merton modelComputation of ExpectationSquare of arithmetic brownian motion processBaxter & Rennie HJM: differentiating Ito integralSimple HJM model, differentiating the bond priceStochastic Leibniz ruleStochastic differential equation of a Brownian MotionHow to calculate the product of forward rates with different reset times using Ito's lemma?For an Ito Process, $dlnX neq fracdXX$ and $(dlnX)^2 = (fracdXX)^2$, but $dlnX neq pm fracdXX$
Yu-Gi-Oh cards in Python 3
Towers in the ocean; How deep can they be built?
New carbon wheel brake pads after use on aluminum wheel?
Why is the US ranked as #45 in Press Freedom ratings, despite its extremely permissive free speech laws?
Regression vs Random Forest - Combination of features
If Nick Fury and Coulson already knew about aliens (Kree and Skrull) why did they wait until Thor's appearance to start making weapons?
What does "shotgun unity" refer to here in this sentence?
Easy to read palindrome checker
Would a grinding machine be a simple and workable propulsion system for an interplanetary spacecraft?
Pulling the principal components out of a DimensionReducerFunction?
How to avoid supervisors with prejudiced views?
Can I use the word “Senior” as part of a job title directly in German?
AB diagonalizable then BA also diagonalizable
From jafe to El-Guest
what's the use of '% to gdp' type of variables?
Can this note be analyzed as a non-chord tone?
"Eavesdropping" vs "Listen in on"
Expressing the idea of having a very busy time
Is there a way to save my career from absolute disaster?
Lucky Feat: How can "more than one creature spend a luck point to influence the outcome of a roll"?
Is French Guiana a (hard) EU border?
How to properly draw diagonal line while using multicolumn inside tabular environment?
How do I fit a non linear curve?
Prepend last line of stdin to entire stdin
Expectation in a stochastic differential equation
The Next CEO of Stack OverflowWhat is Ito's lemma used for in quantitative finance?Question about the stochastic differential equation in the Merton modelComputation of ExpectationSquare of arithmetic brownian motion processBaxter & Rennie HJM: differentiating Ito integralSimple HJM model, differentiating the bond priceStochastic Leibniz ruleStochastic differential equation of a Brownian MotionHow to calculate the product of forward rates with different reset times using Ito's lemma?For an Ito Process, $dlnX neq fracdXX$ and $(dlnX)^2 = (fracdXX)^2$, but $dlnX neq pm fracdXX$
$begingroup$
I'm new to stochastic calculus, I want to find the mean of $X_2$ with $X_t = exp(W_t)$, with $W_t$ a Wiener process.
I used Ito's Lemma is arrive at the SDE:
beginalign
d(X_t) = frac12X_t dt + X_t dW_t
endalign
But how can I get the mean of $X_2$?
itos-lemma sde
asked 2 hours ago
VictorVictor
614
$endgroup$
add a comment |
$begingroup$
I'm new to stochastic calculus, I want to find the mean of $X_2$ with $X_t = exp(W_t)$, with $W_t$ a Wiener process.
I used Ito's Lemma is arrive at the SDE:
beginalign
d(X_t) = frac12X_t dt + X_t dW_t
endalign
But how can I get the mean of $X_2$?
itos-lemma sde
asked 2 hours ago
VictorVictor
614
$endgroup$
add a comment |
$begingroup$
I'm new to stochastic calculus, I want to find the mean of $X_2$ with $X_t = exp(W_t)$, with $W_t$ a Wiener process.
I used Ito's Lemma is arrive at the SDE:
beginalign
d(X_t) = frac12X_t dt + X_t dW_t
endalign
But how can I get the mean of $X_2$?
itos-lemma sde
asked 2 hours ago
VictorVictor
614
$endgroup$
I'm new to stochastic calculus, I want to find the mean of $X_2$ with $X_t = exp(W_t)$, with $W_t$ a Wiener process.
I used Ito's Lemma is arrive at the SDE:
beginalign
d(X_t) = frac12X_t dt + X_t dW_t
endalign
But how can I get the mean of $X_2$?
itos-lemma sde
itos-lemma sde
asked 2 hours ago
VictorVictor
614
asked 2 hours ago
VictorVictor
614
edited 1 hour ago
Victor
asked 2 hours ago
VictorVictor
614
asked 2 hours ago
VictorVictor
614
asked 2 hours ago
VictorVictor
614
614
add a comment |
add a comment |
1 Answer
1
active
oldest
votes
$begingroup$
Assuming you are talking about unconditional expectation, in general you have
$$
mathbbE[X_t] = mathbbE[e^W_t] = e^mathbbE[W_t] + frac12textVar(W_t)
$$
which yields
$$
mathbbE[X_t]= e^frac12 t
$$
Hence,
$$ mathbbE[X_2]= e $$
answered 1 hour ago
RafaelCRafaelC
1263
New contributor
RafaelC is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
$endgroup$
$begingroup$
I'm quite new to the theory. What is the name of the first equality and under which hypotheses is it true?
$endgroup$
– Victor
1 hour ago
1
$begingroup$
@Victor the first equality comes from the moment-generating function of a normal. Take a look here for more details. In general, $mathbbE[e^X] = e^mu + frac12 sigma^2$ holds whenever $X sim mathcalN(mu, sigma^2)$
$endgroup$
– RafaelC
1 hour ago
add a comment |
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: "204"
;
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: false,
noModals: true,
showLowRepImageUploadWarning: true,
reputationToPostImages: null,
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%2fquant.stackexchange.com%2fquestions%2f44854%2fexpectation-in-a-stochastic-differential-equation%23new-answer', 'question_page');
);
Post as a guest
Required, but never shown
1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
$begingroup$
Assuming you are talking about unconditional expectation, in general you have
$$
mathbbE[X_t] = mathbbE[e^W_t] = e^mathbbE[W_t] + frac12textVar(W_t)
$$
which yields
$$
mathbbE[X_t]= e^frac12 t
$$
Hence,
$$ mathbbE[X_2]= e $$
answered 1 hour ago
RafaelCRafaelC
1263
New contributor
RafaelC is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
$endgroup$
$begingroup$
I'm quite new to the theory. What is the name of the first equality and under which hypotheses is it true?
$endgroup$
– Victor
1 hour ago
1
$begingroup$
@Victor the first equality comes from the moment-generating function of a normal. Take a look here for more details. In general, $mathbbE[e^X] = e^mu + frac12 sigma^2$ holds whenever $X sim mathcalN(mu, sigma^2)$
$endgroup$
– RafaelC
1 hour ago
add a comment |
$begingroup$
Assuming you are talking about unconditional expectation, in general you have
$$
mathbbE[X_t] = mathbbE[e^W_t] = e^mathbbE[W_t] + frac12textVar(W_t)
$$
which yields
$$
mathbbE[X_t]= e^frac12 t
$$
Hence,
$$ mathbbE[X_2]= e $$
answered 1 hour ago
RafaelCRafaelC
1263
New contributor
RafaelC is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
$endgroup$
$begingroup$
I'm quite new to the theory. What is the name of the first equality and under which hypotheses is it true?
$endgroup$
– Victor
1 hour ago
1
$begingroup$
@Victor the first equality comes from the moment-generating function of a normal. Take a look here for more details. In general, $mathbbE[e^X] = e^mu + frac12 sigma^2$ holds whenever $X sim mathcalN(mu, sigma^2)$
$endgroup$
– RafaelC
1 hour ago
add a comment |
$begingroup$
Assuming you are talking about unconditional expectation, in general you have
$$
mathbbE[X_t] = mathbbE[e^W_t] = e^mathbbE[W_t] + frac12textVar(W_t)
$$
which yields
$$
mathbbE[X_t]= e^frac12 t
$$
Hence,
$$ mathbbE[X_2]= e $$
answered 1 hour ago
RafaelCRafaelC
1263
New contributor
RafaelC is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
$endgroup$
Assuming you are talking about unconditional expectation, in general you have
$$
mathbbE[X_t] = mathbbE[e^W_t] = e^mathbbE[W_t] + frac12textVar(W_t)
$$
which yields
$$
mathbbE[X_t]= e^frac12 t
$$
Hence,
$$ mathbbE[X_2]= e $$
answered 1 hour ago
RafaelCRafaelC
1263
New contributor
RafaelC is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
answered 1 hour ago
RafaelCRafaelC
1263
New contributor
RafaelC is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
answered 1 hour ago
RafaelCRafaelC
1263
answered 1 hour ago
RafaelCRafaelC
1263
1263
New contributor
RafaelC is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
New contributor
RafaelC is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
RafaelC is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
$begingroup$
I'm quite new to the theory. What is the name of the first equality and under which hypotheses is it true?
$endgroup$
– Victor
1 hour ago
1
$begingroup$
@Victor the first equality comes from the moment-generating function of a normal. Take a look here for more details. In general, $mathbbE[e^X] = e^mu + frac12 sigma^2$ holds whenever $X sim mathcalN(mu, sigma^2)$
$endgroup$
– RafaelC
1 hour ago
add a comment |
$begingroup$
I'm quite new to the theory. What is the name of the first equality and under which hypotheses is it true?
$endgroup$
– Victor
1 hour ago
1
$begingroup$
@Victor the first equality comes from the moment-generating function of a normal. Take a look here for more details. In general, $mathbbE[e^X] = e^mu + frac12 sigma^2$ holds whenever $X sim mathcalN(mu, sigma^2)$
$endgroup$
– RafaelC
1 hour ago
$begingroup$
I'm quite new to the theory. What is the name of the first equality and under which hypotheses is it true?
$endgroup$
– Victor
1 hour ago
$begingroup$
I'm quite new to the theory. What is the name of the first equality and under which hypotheses is it true?
$endgroup$
– Victor
1 hour ago
1
1
$begingroup$
@Victor the first equality comes from the moment-generating function of a normal. Take a look here for more details. In general, $mathbbE[e^X] = e^mu + frac12 sigma^2$ holds whenever $X sim mathcalN(mu, sigma^2)$
$endgroup$
– RafaelC
1 hour ago
$begingroup$
@Victor the first equality comes from the moment-generating function of a normal. Take a look here for more details. In general, $mathbbE[e^X] = e^mu + frac12 sigma^2$ holds whenever $X sim mathcalN(mu, sigma^2)$
$endgroup$
– RafaelC
1 hour ago
add a comment |
Thanks for contributing an answer to Quantitative Finance 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%2fquant.stackexchange.com%2fquestions%2f44854%2fexpectation-in-a-stochastic-differential-equation%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
