Variance of Stochastic Integral $int_0^1t^2 dWt$Expected value of the stochastic integral $int_0^t e^as dW_s$Expectation and variance of this stochastic processVariance of Ito IntegralFind a process $f=f(t,W_t)$ such that another process is a martingaleStochastic Calculus - Ito decompositionIto formula - some doubtsDistribution of stochastic integralWhat are the components in the Ito Process?Representation of Ito integralCalculating Variance of the stochastic process

How did Arya survive the stabbing?

How to safely derail a train during transit?

Why escape if the_content isnt?

How do I find the solutions of the following equation?

How to check is there any negative term in a large list?

How does Loki do this?

Are student evaluations of teaching assistants read by others in the faculty?

How to be diplomatic in refusing to write code that breaches the privacy of our users

How did Doctor Strange see the winning outcome in Avengers: Infinity War?

Why not increase contact surface when reentering the atmosphere?

What is the difference between "behavior" and "behaviour"?

Tiptoe or tiphoof? Adjusting words to better fit fantasy races

Anatomically Correct Strange Women In Ponds Distributing Swords

Nautlius: add mouse right-click action to compute MD5 sum

CREATE opcode: what does it really do?

Do sorcerers' subtle spells require a skill check to be unseen?

Applicability of Single Responsibility Principle

Lay out the Carpet

Why Were Madagascar and New Zealand Discovered So Late?

Was Spock the First Vulcan in Starfleet?

How to escape string to filename? It is in backup a file append date

Gears on left are inverse to gears on right?

Purchasing a ticket for someone else in another country?

System.debug(JSON.Serialize(o)) Not longer shows full string



Variance of Stochastic Integral $int_0^1t^2 dWt$


Expected value of the stochastic integral $int_0^t e^as dW_s$Expectation and variance of this stochastic processVariance of Ito IntegralFind a process $f=f(t,W_t)$ such that another process is a martingaleStochastic Calculus - Ito decompositionIto formula - some doubtsDistribution of stochastic integralWhat are the components in the Ito Process?Representation of Ito integralCalculating Variance of the stochastic process













0












$begingroup$


I want to find Variance of integral $int_0^1t^2 dWt$



$W_t$ is Brownian motion



What I did:



I used Ito formula and got:



$int_0^1t^2 dWt = W_1 - int_0^1 2tW_t dt$ (correct me if I'm wrong)



I do not know how to compute $int_0^1 2tW_t dt$ or compute variance for whole answer.






stochastic-processes stochastic-calculus stochastic-integrals







share|cite|improve this question











$endgroup$







  • 2




    $begingroup$
    by Ito the variance is $int_0^1 t^4 dt$
    $endgroup$
    – Math-fun
    yesterday















0












$begingroup$


I want to find Variance of integral $int_0^1t^2 dWt$



$W_t$ is Brownian motion



What I did:



I used Ito formula and got:



$int_0^1t^2 dWt = W_1 - int_0^1 2tW_t dt$ (correct me if I'm wrong)



I do not know how to compute $int_0^1 2tW_t dt$ or compute variance for whole answer.






stochastic-processes stochastic-calculus stochastic-integrals







share|cite|improve this question











$endgroup$







  • 2




    $begingroup$
    by Ito the variance is $int_0^1 t^4 dt$
    $endgroup$
    – Math-fun
    yesterday













0












0








0





$begingroup$


I want to find Variance of integral $int_0^1t^2 dWt$



$W_t$ is Brownian motion



What I did:



I used Ito formula and got:



$int_0^1t^2 dWt = W_1 - int_0^1 2tW_t dt$ (correct me if I'm wrong)



I do not know how to compute $int_0^1 2tW_t dt$ or compute variance for whole answer.






stochastic-processes stochastic-calculus stochastic-integrals







share|cite|improve this question











$endgroup$




I want to find Variance of integral $int_0^1t^2 dWt$



$W_t$ is Brownian motion



What I did:



I used Ito formula and got:



$int_0^1t^2 dWt = W_1 - int_0^1 2tW_t dt$ (correct me if I'm wrong)



I do not know how to compute $int_0^1 2tW_t dt$ or compute variance for whole answer.






stochastic-processes stochastic-calculus stochastic-integrals




stochastic-processes stochastic-calculus stochastic-integrals






share|cite|improve this question















share|cite|improve this question













share|cite|improve this question




share|cite|improve this question








edited 10 hours ago







mahdi

















asked yesterday









mahdimahdi

7412




7412







  • 2




    $begingroup$
    by Ito the variance is $int_0^1 t^4 dt$
    $endgroup$
    – Math-fun
    yesterday












  • 2




    $begingroup$
    by Ito the variance is $int_0^1 t^4 dt$
    $endgroup$
    – Math-fun
    yesterday







2




2




$begingroup$
by Ito the variance is $int_0^1 t^4 dt$
$endgroup$
– Math-fun
yesterday




$begingroup$
by Ito the variance is $int_0^1 t^4 dt$
$endgroup$
– Math-fun
yesterday










1 Answer
1






active

oldest

votes


















3












$begingroup$

Expectation of Ito integral of a deterministic function wrt to a Brownian motion is $0$. Thus,



$$ mathbbVarleft[int_0^1 t^2 d W_t right]= mathbbEleft[left(int_0^1 t^2 d W_tright)^2right]$$



Applying Ito Isometry, we obtain



$$ mathbbEleft[left(int_0^1 t^2 d W_tright)^2right] = mathbbEleft[int_0^1 t^4 dtright] = int_0^1 t^4 dt = frac15 $$






share|cite|improve this answer









$endgroup$












    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
    );



    );













    draft saved

    draft discarded


















    StackExchange.ready(
    function ()
    StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f3163682%2fvariance-of-stochastic-integral-int-01t2-dwt%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









    3












    $begingroup$

    Expectation of Ito integral of a deterministic function wrt to a Brownian motion is $0$. Thus,



    $$ mathbbVarleft[int_0^1 t^2 d W_t right]= mathbbEleft[left(int_0^1 t^2 d W_tright)^2right]$$



    Applying Ito Isometry, we obtain



    $$ mathbbEleft[left(int_0^1 t^2 d W_tright)^2right] = mathbbEleft[int_0^1 t^4 dtright] = int_0^1 t^4 dt = frac15 $$






    share|cite|improve this answer









    $endgroup$

















      3












      $begingroup$

      Expectation of Ito integral of a deterministic function wrt to a Brownian motion is $0$. Thus,



      $$ mathbbVarleft[int_0^1 t^2 d W_t right]= mathbbEleft[left(int_0^1 t^2 d W_tright)^2right]$$



      Applying Ito Isometry, we obtain



      $$ mathbbEleft[left(int_0^1 t^2 d W_tright)^2right] = mathbbEleft[int_0^1 t^4 dtright] = int_0^1 t^4 dt = frac15 $$






      share|cite|improve this answer









      $endgroup$















        3












        3








        3





        $begingroup$

        Expectation of Ito integral of a deterministic function wrt to a Brownian motion is $0$. Thus,



        $$ mathbbVarleft[int_0^1 t^2 d W_t right]= mathbbEleft[left(int_0^1 t^2 d W_tright)^2right]$$



        Applying Ito Isometry, we obtain



        $$ mathbbEleft[left(int_0^1 t^2 d W_tright)^2right] = mathbbEleft[int_0^1 t^4 dtright] = int_0^1 t^4 dt = frac15 $$






        share|cite|improve this answer









        $endgroup$



        Expectation of Ito integral of a deterministic function wrt to a Brownian motion is $0$. Thus,



        $$ mathbbVarleft[int_0^1 t^2 d W_t right]= mathbbEleft[left(int_0^1 t^2 d W_tright)^2right]$$



        Applying Ito Isometry, we obtain



        $$ mathbbEleft[left(int_0^1 t^2 d W_tright)^2right] = mathbbEleft[int_0^1 t^4 dtright] = int_0^1 t^4 dt = frac15 $$







        share|cite|improve this answer












        share|cite|improve this answer



        share|cite|improve this answer










        answered 15 hours ago









        MdocMdoc

        556515




        556515



























            draft saved

            draft discarded
















































            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.




            draft saved


            draft discarded














            StackExchange.ready(
            function ()
            StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f3163682%2fvariance-of-stochastic-integral-int-01t2-dwt%23new-answer', 'question_page');

            );

            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







            -

            Popular posts from this blog

            Trouble understanding the speech of overseas colleaguesHow can I better understand manager or clients with strong accents?Adding more movement and speech at the fundamental level to a highly-sedentary job?Difficulty in understanding Manager's accent(language and communication)How to adjust yourself where your colleagues are not understanding to you?Understanding manager's expectationsForeigner and colleagues using slangHaving difficulty understanding meetingsHow do you breathe when giving a speech?Trouble Waking Up for Emergencies (On-Call)Problems with colleaguesColleagues feeling insecure when I do my work

            Image
            Number of words that can be made using all the letters of the word W, if Os as well as Is are separated is? Class Action - which options I have? Do the temporary hit points from the Battlerager barbarian's Reckless Abandon stack if I make multiple attacks on my turn? Implement the Thanos sorting algorithm Invade the Pyramid if you Dare Lay out the Carpet How can I get through very long and very dry, but also very useful technical documents when learning a new tool? A Rare Riley Riddle Why Were Madagascar and New Zealand Discovered So Late? How do I extract a value from a time formatted value in excel? Inappropriate reference requests from Journal reviewers How can I quit an app using Terminal? How do we know the LHC results are robust? Escape a mm/dd/YY backup date in a file name Why escape if the_content isnt? Do sorcerers' Subtle Spells require a skill check to be unseen? Sort a list by elements of another list What does 算不上 mean ...
            Read more

            Is the concept of a “numerable” fiber bundle really useful or an empty generalization?Non trivial vector bundle over non-paracompact contractible spaceExample of fiber bundle that is not a fibrationGlobalising fibrations by schedulesFiber bundle = principal bundle + fiber?Numerable covers from the point of view of Grothendieck topologiesGlobal sections for torus fiber bundleAre there analogs of smooth partitions of unity and good open covers for PL-manifolds?Two natural maps asssociated with the nerve of a coverDescent theory, fibrations, and bundlesIn which sense are Euler-Lagrange PDE's on fiber bundles quasi-linear?What is the local structure of a fibration?Complete proof of Homotopy invariance of a numerable fiber bundle based on CHPLocally trivial fibration over a suspension

            Image
            Is the concept of a “numerable” fiber bundle really useful or an empty generalization? Non trivial vector bundle over non-paracompact contractible spaceExample of fiber bundle that is not a fibrationGlobalising fibrations by schedulesFiber bundle = principal bundle + fiber?Numerable covers from the point of view of Grothendieck topologiesGlobal sections for torus fiber bundleAre there analogs of smooth partitions of unity and good open covers for PL-manifolds?Two natural maps asssociated with the nerve of a coverDescent theory, fibrations, and bundlesIn which sense are Euler-Lagrange PDE's on fiber bundles quasi-linear?What is the local structure of a fibration?Complete proof of Homotopy invariance of a numerable fiber bundle based on CHPLocally trivial fibration over a suspension 13 4 $begingroup$ Numerable fiber bundles are defined by Dold (DOLD 1962 - Partitions of Unity in theory of Fibrations) as a generalization of fiber bundles over a paracompact space...
            Read more

            Are student evaluations of teaching assistants read by others in the faculty?Do teaching evaluations lead to lower standards in class?How can one learn from poor teaching evaluations?From which level of education, is it better to choose teacher assistants?Does better learning affect teaching evaluations? If so, how?Alternatives to relying solely on student evaluations of teaching to help administrators more accurately assess teaching effectivenessHow to handle communication about the class outside of the official channels, when I am a TA for a class that a friend is taking?What should I do as a TA if the instructor is not teaching properly?Are teaching assistants responsible for grading assignments not in their course?My students compliment my teaching and say they now approach the subject with newfound enthusiasm. Can I ask them to put this in writing?How to get teaching evaluations for a course as a TA / Co-instructor

            Image
            Was Spock the First Vulcan in Starfleet? India just shot down a satellite from the ground. At what altitude range is the resulting debris field? Lay out the Carpet Is oxalic acid dihydrate considered a primary acid standard in analytical chemistry? Is a stroke of luck acceptable after a series of unfavorable events? Tiptoe or tiphoof? Adjusting words to better fit fantasy races How did Doctor Strange see the winning outcome in Avengers: Infinity War? Is there a problem with hiding "forgot password" until it's needed? How does it work when somebody invests in my business? Trouble understanding the speech of overseas colleagues How to write papers efficiently when English isn't my first language? A particular customize with green line and letters for subfloat How do scammers retract money, while you can’t? How to safely derail a train during transit? Unreliable Magic - Is it worth it? Why not increase contact surface when reentering the atm...
            Read more